Title: Geometric Algebra, Geometric Calculus
Post by: kram1032 on May 15, 2013, 02:44:44 AM
Today -- (looking at the clock) -- yesterday, I found out about Geometric Algebra and Geometric Calculus. It is essentially Vector Algebra and - Calculus combined with how Clifford Algebras work.
That way, pretty much any operations you can think of are well defined.
It features three kinds of product (called the "vector" product which is equivalent to, say a complex or quaternion product, the "wedge" or "outer" product, which is equivalent to a cross-product and the "inner" product which is like the dot product) that are all defined in a very uniform, beautiful way across all types of vectors and scalars. Furthermore, stuff like)
make sense in these. - You can add scalars to vectors and it's all well defined.
Turns out, defining vectors in this way gives rise to absolutely beautiful simplifications and reveals structures, "classical" ways of doing this hide in notational clutter.
Another plus is, that these are absolutely dimensionally independend. For instance, the cross-product, as you probably know it, is limited to 3 Dimensions. For 2 Dimensions, you have to apply a "hack" by embedding your 2D space into a 3D space. For 4 Dimensions and up, you have to do things differently and essentially hack your way to a proper definition. In this, essentially, rewrite of Vector Algebra, all that just goes away and uniformly applies across all dimensions.
Now if you were careful, you noticed that I mentioned dimensional independence and clifford algebras. And if you delved into this for a bit, that clifford algebras usually are associated with complex numbers, quaternions, octonions and so on.
What this extension does is the "mathematically sound" way of having algebras over ANY number of dimensions. - And that in a way that's much more compact, simpler and more beautiful than any other I've seen before.
It's said to have a steep learning-curve but the reason for that is mostly that you have to throw over board a lot of old practices if you worked with vectors rigorously before. You might actually have a slightly easier time if you didn't do so as much.
Furthermore, there is Geometric Calculus. This is where you can delve into spacetime and other manifolds, etc. Classically you'd need quite convoluted-looking Tensor calculus for this to properly work out. Geometric Calculus looks much cleaner and in some ways is actually more powerful than that. If you can get yourself to work through what is known so far, a lot of problems will suddenly seem a lot simpler to you.
Here you'll find introductions of how it all would benefit physics. (Being a physics student, I can relate very much)
http://geocalc.clas.asu.edu/
http://geocalc.clas.asu.edu/html/UGC.html
And here is a rather computer-science oriented collection of explanations (which might interest those of you more who write their own renderers and stuff):
Both angles are very valuable and I'm sure, if you get into it a bit, you'll see the advantages too.
Here is another explanation: http://www.mrao.cam.ac.uk/~clifford/introduction/intro/node2.html
As one mostly visual show-case of how much simpler things become when using this formalism, look at the following image:
It presents four different ways of saying the same thing. The first one, Vector calculus, only works for euclidean space and can't deal with relativism. It's also the least compact one and looks the most complex.
Tensor calculus, the second one works for flat relativistic settings. They are rather hard to wrap your head around.
Differential Forms are very abstract. You can already see that things finally become simpler, and additionally it can all be applied to arbitrary space-times.
Geometric Calculus, as the name implies, from ground up has very visual, geometric interpretations. Furthermore, it manages to capture the entirety of Maxwell's Equations in just a single, short equation. It just can not get simpler than that. And because the interpretations are rather visual, the meaning of all those parts is also much easier to conceptualize than in any other form. Despite the generality.
(http://puu.sh/2TUyj.png)
(the "^" in those is one kind of multiplication, the "." another and when you leave out a symbol, it's the one that is exactly like complex or quaternion multiplication. "^" is essentially your "imaginary" part of that same multiplication but its meaning typically is something like space or momentum, a vector-quantity, while "." gives you the "real" part, a scalar, representing something like Energy or time. That way you can have space-time or energy-momentum in one concept, ultimately making all those formulae a single line. - the top line of that block)
That way, pretty much any operations you can think of are well defined.
It features three kinds of product (called the "vector" product which is equivalent to, say a complex or quaternion product, the "wedge" or "outer" product, which is equivalent to a cross-product and the "inner" product which is like the dot product) that are all defined in a very uniform, beautiful way across all types of vectors and scalars. Furthermore, stuff like
Turns out, defining vectors in this way gives rise to absolutely beautiful simplifications and reveals structures, "classical" ways of doing this hide in notational clutter.
Another plus is, that these are absolutely dimensionally independend. For instance, the cross-product, as you probably know it, is limited to 3 Dimensions. For 2 Dimensions, you have to apply a "hack" by embedding your 2D space into a 3D space. For 4 Dimensions and up, you have to do things differently and essentially hack your way to a proper definition. In this, essentially, rewrite of Vector Algebra, all that just goes away and uniformly applies across all dimensions.
Now if you were careful, you noticed that I mentioned dimensional independence and clifford algebras. And if you delved into this for a bit, that clifford algebras usually are associated with complex numbers, quaternions, octonions and so on.
What this extension does is the "mathematically sound" way of having algebras over ANY number of dimensions. - And that in a way that's much more compact, simpler and more beautiful than any other I've seen before.
It's said to have a steep learning-curve but the reason for that is mostly that you have to throw over board a lot of old practices if you worked with vectors rigorously before. You might actually have a slightly easier time if you didn't do so as much.
Furthermore, there is Geometric Calculus. This is where you can delve into spacetime and other manifolds, etc. Classically you'd need quite convoluted-looking Tensor calculus for this to properly work out. Geometric Calculus looks much cleaner and in some ways is actually more powerful than that. If you can get yourself to work through what is known so far, a lot of problems will suddenly seem a lot simpler to you.
Here you'll find introductions of how it all would benefit physics. (Being a physics student, I can relate very much)
http://geocalc.clas.asu.edu/
http://geocalc.clas.asu.edu/html/UGC.html
And here is a rather computer-science oriented collection of explanations (which might interest those of you more who write their own renderers and stuff):
| http://www.science.uva.nl/ga/ | (newer page) |
| http://staff.science.uva.nl/~leo/clifford/index.html | (older page) |
Both angles are very valuable and I'm sure, if you get into it a bit, you'll see the advantages too.
Here is another explanation: http://www.mrao.cam.ac.uk/~clifford/introduction/intro/node2.html
As one mostly visual show-case of how much simpler things become when using this formalism, look at the following image:
It presents four different ways of saying the same thing. The first one, Vector calculus, only works for euclidean space and can't deal with relativism. It's also the least compact one and looks the most complex.
Tensor calculus, the second one works for flat relativistic settings. They are rather hard to wrap your head around.
Differential Forms are very abstract. You can already see that things finally become simpler, and additionally it can all be applied to arbitrary space-times.
Geometric Calculus, as the name implies, from ground up has very visual, geometric interpretations. Furthermore, it manages to capture the entirety of Maxwell's Equations in just a single, short equation. It just can not get simpler than that. And because the interpretations are rather visual, the meaning of all those parts is also much easier to conceptualize than in any other form. Despite the generality.
(http://puu.sh/2TUyj.png)
(the "^" in those is one kind of multiplication, the "." another and when you leave out a symbol, it's the one that is exactly like complex or quaternion multiplication. "^" is essentially your "imaginary" part of that same multiplication but its meaning typically is something like space or momentum, a vector-quantity, while "." gives you the "real" part, a scalar, representing something like Energy or time. That way you can have space-time or energy-momentum in one concept, ultimately making all those formulae a single line. - the top line of that block)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on May 15, 2013, 06:26:07 PM
I believe, the study of this would generally be very interesting, but the standard 2-dimensional Manelbrot-Set of this can be written like so:
z0=z02+z12+z22-z32+c0 (- beware of that last "-")
z1=2 z0z1+c1
z2=2 z0z2+c2
z3=2 z0z3+c3
Now you'll be definitely be wondering how this is 2D if I wrote down four equation.
That's because Geometric Algebra is more careful than classical Vector Algebra to differentiate between differernt types of vectors, so if you multiply together two vectors, you'll get a bivector or "2-vector".
In higher dimensions, there are even more constructs, for instance, if you multiply together a bivector and a vector in 3D, you'll get a trivector. In 2D, that doesn't work anymore.
If you multiply together a bivector and a vector, you'll get another normal vector and if you multiply two bi-vectors, you'll get a scalar (a 0-vector).
Additionally, unlike what you have ever learned from vectors, you can add together any type of vector. - This part works just like how you differentiate between real and imaginary part in a complex number or scalar and vectorial part in quaternions.
You could certainly try to look at each of the possible planes or try different cut-planes that aren't the main-planes. But the "classical 2D" plane (which would, however, clearly NOT give you the standard M-set), in this case would be the z1z2-plane.
z0 is a scalar (0-vector) and z3 is a bi-vector (which for 2D classically is refered to as "pseudo-scalar" and incidentally acts just like the imaginary part of a complex number).
z1 and z2 represent the x and y direction of the normal flat 2D-plane.
Now I do not expect these rules to give particularly interesting fractal patterns, and I'm almost certain, this has been tried before (at the very least something very close to it was) but who knows, maybe this leads to some surprising dynamics. Typically though, mathematically rigorous definitions that are efficient, symmetric and nice, give rather boring resuls, so we'll see.
The plane that should be most like the original M-set we all know and love, should be the one that combines the scalar and pseudo-scalar values, so the z0z3-plane.
Indeed, if you just set z1, z2, c1 and c2 to 0, you'll recover the standard, well known complex numbers.
z0=z02+z12+z22-z32+c0 (- beware of that last "-")
z1=2 z0z1+c1
z2=2 z0z2+c2
z3=2 z0z3+c3
Now you'll be definitely be wondering how this is 2D if I wrote down four equation.
That's because Geometric Algebra is more careful than classical Vector Algebra to differentiate between differernt types of vectors, so if you multiply together two vectors, you'll get a bivector or "2-vector".
In higher dimensions, there are even more constructs, for instance, if you multiply together a bivector and a vector in 3D, you'll get a trivector. In 2D, that doesn't work anymore.
If you multiply together a bivector and a vector, you'll get another normal vector and if you multiply two bi-vectors, you'll get a scalar (a 0-vector).
Additionally, unlike what you have ever learned from vectors, you can add together any type of vector. - This part works just like how you differentiate between real and imaginary part in a complex number or scalar and vectorial part in quaternions.
You could certainly try to look at each of the possible planes or try different cut-planes that aren't the main-planes. But the "classical 2D" plane (which would, however, clearly NOT give you the standard M-set), in this case would be the z1z2-plane.
z0 is a scalar (0-vector) and z3 is a bi-vector (which for 2D classically is refered to as "pseudo-scalar" and incidentally acts just like the imaginary part of a complex number).
z1 and z2 represent the x and y direction of the normal flat 2D-plane.
Now I do not expect these rules to give particularly interesting fractal patterns, and I'm almost certain, this has been tried before (at the very least something very close to it was) but who knows, maybe this leads to some surprising dynamics. Typically though, mathematically rigorous definitions that are efficient, symmetric and nice, give rather boring resuls, so we'll see.
The plane that should be most like the original M-set we all know and love, should be the one that combines the scalar and pseudo-scalar values, so the z0z3-plane.
Indeed, if you just set z1, z2, c1 and c2 to 0, you'll recover the standard, well known complex numbers.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: jehovajah on May 17, 2013, 09:42:29 AM
Do not be so fatalistic Kram!
My Newtonian triples are based on these type of notations!
My Newtonian triples are based on these type of notations!
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on May 17, 2013, 06:16:00 PM
That's what I meant when I was saying, stuff very much like this (maybe even exactly this) already was tried.
And thanks for that new word. Never heard it before, lol
And thanks for that new word. Never heard it before, lol
Title: Re: Geometric Algebra, Geometric Calculus: The Algebras of Geometries
Post by: jehovajah on May 22, 2013, 12:11:36 PM
Kram asked if i might write an introduction to the Algebras of Geometries now called geometric algebra.
Please let me know if there is anything that needs clarification.
Please let me know if there is anything that needs clarification.
Quote
In Grassmanns toolkit everything fits together. Strecken as construction lines between points reveal a formal notational structure, that is a Begriffe. This literally means a way to grip things that are as slippery as distinctions and Ideas/ forms. The notation literally can have a one to one onto correspondence with each idea , distinction or form. This means one can represent the essential process of experiencing, distinguishing, comparing, recognising, representing and manipulating the aspects of an actual or imagined interaction with space by a careful selection of symbols.
We do this all the time, so naturally and in several neural representation systems that we forget or ignore it as a fundamental process. We regard it through the smokescreen of language , not realising that our languages are cultural constructions. Thus if we build a crystal clear language or jargon in which every symbol has a precise meaning, we can actually use it to explore and model inner and outer experiences in relation to a form/ Idea. This is precisely and painstakingly what the Grassmanns have done in their Ausdehnungslehre series,
In the course of doing this Hermann revisited every fundamental notion in science, really deconstructing Euclid and resynthesising his Stoikeioon. This was Justus stated goal. Hermann got interested in it at a stage when Justus had implemented school trials of many of his ideas and reformulations, picking out an inner structure that Justus could not apprehend because he was the pathfinder! Finally Robert extended the analysis to the foundation of philosophy itself! He called this approach Formenlehre, and really engaged with the Platonic philosophy or theory of forms and Ideas.
That Euclid should serve to unite a constructive approach to geometry, and a empirical approach to philosophy says to me that the Stoikeioon is not a book about geometry or mathematics. It is a book about the philosophy of Form/ ideas.
Hermann literally was between these 2 seminal path finding researchers. He did not live in the same household as his father and brother, but with his uncle. This was a child adoption arrangement, but not an abandonment by any means. Hermann seems to have always known who his biological father was. But he lived with and was brought up in his uncles family due to infertility it seems.
Hermann's absorption of Euclid was thus mostly unconscious and lifelong. It was not as a classical training in Euclid, but as a result of an innovative programme of restructuring geometrical education that he imbibed Euclidean forms. Much the same happened in my education. When geometry was taught , it was not to understand Euclid it was as a subject called Geometry. It was not until much much later in life that I actually began to look at the Greek text!
What that does is dissociate the student from the originator. The student never actually knows what the original thought was. This is why I read Grassmann's own words, in German or Prussian .
The second point is, by the time you do get to study the original the prejudice has already been imposed, so the original thought is almost indiscernible!
The advantage that those who have a classics education have is that through learning the actual language they can " rinse" their brains of prejudicial ideas. The more languages they learn to meditate in, the clearer their conceits become.. But then they face the problem of dissociation: they no longer think the same as those who are brain washed differently!
Any way. Newton read the original Greek text or a Latin copy. Thus the quality of his thinking and insight was suffused and informed by Platonic ideals. Hamilton read Euclid in the Greek and was similarly motivated. The Grassmanns had access to the notions of geometry associated to Euclid. Justus clearly had experience of parts of the text, but I do not know yet which parts or books.
Hermann was a great linguist and that makes me think he had access to the texts either directly or through deconstructed copies in the form of lexicons. Someway, somehow, the ideas and forms of Euclid suffused all the above mentioned individuals.
Thus when Newton established his " vector" algebra based on the parallelogram , it was not in isolation. Hooke, Huygens and all the geometers knew precisely what he meant. But when Grassmann similarly deconstructed and resynthesized the geometry of the parallelogram, suddenly he is talking " obscurely"?
The only difference between Hermann and Newton is the subject boundary called algebra. In newtons day algebra was not a subject, it was included within general Rhetoric and Reasoning. Reason is derive from ratio and proportion, logos and Kairos in Greek, and thus not considered part of the mathematical subject boundary. It was and still is within the Philosophy subject boundary.
Thus we have this academic compartmental approach blocking understanding for millennia! The shake up and mixing of the Humboldt reforms , creating the mix of primary and higher academic teachers , alongside the philosophical argument about the structure and nature of mathematics and the sciences engendered a school of thought that knowledge is constructed, not discoursed! Discoursed means several things all at the same time: but the essential idea is that you have to run round all over the place discovering through inquiry, discussion and debate what may be called divine or spiritual Truths, which you intuitively know to be true!
The constructionists basically say, truth is not the criteria, fact is. It is true because it is fact, that is because it has been constructed!
Thus Justus enters the fray , utilising the growing conception of ring and group theory as expounded in crystallography. He actually makes a fundamental contribution to Verbindungslehre as it was called before it came to be dominated by others who renamed it Kôrperlehre!.
To get to the point , an analytical approach to geometry in order to reconstruct it on a systematic, logical and congruent basis revealed a repeated combinatorial structure in every geometry: the geometry of the line, the geometry of the plane, and in the geometry of the Raume , that is 3d geometry. This structure was deliberately based on Arithmetic, because it was believed arithmetic was logically pure and unsullied.
Justus analysis showed however that this was not the case, and he made several suggestions at how to put it right. Some of which were conceptually confused. He could not resolve certain logical difficulties without bringing in the observer as a crucial part. This actually meant that general rules were subject to the individual assenting to a consensus. In those days they believed that the consensus could get it right , even though they were agitating against an old consensus which they felt was wrong!
So Grassmann J, H and R were looking at the ring or group structure of various geometries and finding connections between them. When Hermann looked at the triangle he clearly picked out an additive Algebra! He was looking at geometry and he could see an Algebra. It was only manifest when the correct Notation was used.
Hmm.. Interesting but not earth shattering, until he noticed the same thing when he was looking at the Geometry of the Quadrilaterals. Again, looking at geometry , with the appropriate Notation revealed an Algebra, but this time it was multiplicative! And in addition it connected with the triangular algebra of addition to produce a distributive rule of combination!
This was so intriguing that he began to explore it and found that the " Algebra" held true. Testing it a bit harder he put in the metrics of length and it still held true. It was when he put in the notion of direction that his world turned upside down! The factors , if you could call them that in the analogy did not commute, but instead required the sign that denoted direction to change when they change position.
After waiting some time to get over the shock and general unease at his conclusions he tested them over and over and found them to be logically consistent. He then decided to devote his life's work to exploring these geometric Algebras. There was much work to be done, many gaps in the algebras to explore, but his hard work and dedication to detail, following the strict guidelines of his father seemed to be being rewarded handsomely. He entered and won mathematical competitions to the astonishment of all around him! He found independent confirmation of his ideas in other researchers work. He read and digested Lagrangian Celestial. Mechanics with astonishing ease because his insight into the geometrical algebra suddenly made it simpler clearer, more symmetrical and beautiful!
In tackling the problem of Ebb and Flow he discovered not only the nascent hyperbolic geometry, but his insight revealed its fundamental algebra. The algebra of Newtonian vectors, as understood by Lagrange was suddenly placed before him, and he could clearly see the parallelogram and in this case the rectangular parallelogram geometric algebra.
Newton and thus Lagrange had fully worked this algebra out. In fact most researchers like Huygens, Leibniz Hooke for example were fully conversant with it. But it was called " algebra" only loosely. It was to Newton his own private cogitation by which he mentally manipulated ideas and relationships to gain insights and find solutions. Although Newton was highly organised and structured in his thinking, he did not see that that was important enough to publish.
Bombelli is probably the first author in this epoch to write a book mostly about Algebra. Descartes is the next author,of renown, but he called it Geometry. De Fermat popularised this geometric algebra, but it was Wallis that, drawing upon Euclid and Barrow wrote the first real modern book on Cartesian Geometric Algebra with his great insight. He pestered Newton for his algebraic notations, because he believed that through studying the algebraic reasonings of genius, students could benefit and emulate and surpass. Thus Wallis's work was the standard for Algebra for a long time. And it was always a geometric algebra.
The Grassmanns were different. They studied the Algebras of the geometries, not the algebra of the geometers!
The algebra of the geometries as I explained above required appropriate notation or terminology to distinguish. Thus Hermann struggled to find for each geometry that appropriate notation that made its appearance manifest, or clear, visible(Anschaung). This was like making ghosts or spirits visible. It was like making subjective notions, ideas and forms visible. It was giving form to an invisible structure of formal thinking , revealing how it followed similar and analogous patterns in all the geometries.
So when Hermann came upon the projective geometries, especially the Newtonian decomposition of " vectors" as forces or velocities, he recognised that it was an algebra that applied like an algorithm to any description of physical situations.. He might have petered out at this stage, because essentially he was going to be repeating Newtons Classic Principia. He would have brought little that was new to the discussion. However, his strong notion of geometric algebra in a given geometry lead him to look for addition, multiplication analogues in the projective geometry. He was able to bypass the detail and see the product, and how the parallelogram formed the general product( product here means the constructed form, which Hermann long ago had convinced himself was an analogue of multiplication). But then he found the inner product and with it division!
In the parallelogram geometry Hermann had been able to see that addition and multiplication algebras existed. He therefore knew that if he could solve for the parallelogram he could solve any problem that could be reduced to its form. However, because there was no sense of division, the algebra of the parallelogram geometry was incomplete. In fact it was full of holes.
Herrmann was already considerably blessed with simplifications due to his earlier discoveries. His reasoning and equation formulation was already considerably shorter and smoother than his contemporaries, simply by using this geometric algebra through appropriate notation as his page layout. What I mean by this is that mathematical notation is set out on the page. If you organise it carefully you can make beneficial use of that layout. To speed up and streamline calculation.
The organisation of the calculation on the page has always been a pedagogical concern. It was clear that " neatness" helped in solving problems and performing calculations accurately.rows and columns have always been a significant part of the mathematical discipline since Babylonian script was invented and tablets of information recorded. But the crucial geometric structure of a page layout is derived not from cuneiform, but from Mosaics or Arithmoi.
An Arithmos is not just a mosaic it is the fundamental of geometry itself! As a fundamental of geometry it is used to record Astrology( astro-metry,astro-nomy astro-logia). Thus these mosaics become fundamental and obscured organising principles. Onto such a mosaic a geometrical form may be drawn, the mosaic then representing an epipedos epiphaneia, a so calle flat surface, but in fact often a very colourful abstract art form we now call a mosaic.
"On such a platform geometry can be done !" mused Pythagoras. And of course he was right. Geometry is always done on an embedded mosaic. Because we have lost sight of this we do not understand Arithmos, Arithmoi, and how forms can be shown to be equal without some other notion like length or area or volume. For Greek mathematics, these fundamental notions are embedded in the mosaic. They are the literal structure of space .
So the layout on the page was also in Hermanns mind and when he discovered the Type he called the inner product he states" but this notation also places another product ( to the exterior product as he soon calls it) TO THE side! ( zur Seite), by this I think he means that alongside the exterior product one must also, on the page work out the interior product. The reason is that when both these are done they are a proportion or ratio( by dividing one into the other a fraction) that has a valid value. This value he realised uniquely replaces the angle in parallelogram geometry and provides proportion,ratio and division into the algebraic tool box!
He knew this to be true because he had done the work in his Ebb and Flow paper In which he had used the hyperbolic sin and cosine to create a single multiple form involving a Strecken and an exponential function.
At the time he had taken it as evidence of the parallelogram multiplicative distribution rule, what I called the law of 3 Strecken. But now his insight into the inner product made him realise it was based on the projected Strecken ( obviously!) and not just any Strecken and the angle between them. The angle was not the important ratio of magnitude . A more general ratio of magnitude was involved and that was that of the outer and inner product!
Angle has long been one of those unquestioned measuring algorithms. But it needed to be questioned, and Newton, Cotes and DeMoivre did question it, particularly as it did not concur with astronomical practice. Astronomers use arcs and always have. Someone, some teacher converted the arc into the angle notion and created a serious problem. Or rather later generations misread the symbol for arc and mistakenly developed the notion of angle. Cotes apparently suggested the radian as an arc measure which could easily be used astronomically and geometrically for mundane land measure. The Greeks used Chords, not angles. Each chord has an associated arc, and the 2 together form the connecting link to orbital motions in the heavens. Thus the straight line measure of spread is precisely what the sine and cosine tables record in ratio form. We give a precise ratio of cord to diameter to measure a circular arc. We may never be able to measure the arc directly but we can construct it and through that solve for the triangle AND for the arc by approximation..
Grassmann had now got a complete ring in his parallelogram Algebra. But more than that he could see how to generalise it to any form made up of facets like a crystal.. This is what he means by n dimensional Algebra! Any complex form made up of parallelograms is n dimensional, depending on how many facets were distinguishable. Each of these facets he called a space, and worked out, purely using his algebraic toolbox for parallelograms how to construct a crystal, and how to distinguish crystal forms.
This is where he found and corrected a mistake made by his father. The fact that he could see it clearly was testament to the powerful tools he had created for describing the Algebra of geometries!
There was yet more research to do, but he wanted to publish his first volume and his results to create a stir and get others involved in the research which was now an extensive and rich field of study. That plan and hope backfired on him disastrously, and almost extinguished his belief in his system. That is why when his brother offered , demanded that he redact and republish his work, insisting that it was not to be left idle!, he was willing to give in to his brother's views of his work. Later, much encouraged by the response he reasserts his own view republishing his unredacted original with annotations.
This is the background to the Grassmanns contribution to a revolution in science.
We do this all the time, so naturally and in several neural representation systems that we forget or ignore it as a fundamental process. We regard it through the smokescreen of language , not realising that our languages are cultural constructions. Thus if we build a crystal clear language or jargon in which every symbol has a precise meaning, we can actually use it to explore and model inner and outer experiences in relation to a form/ Idea. This is precisely and painstakingly what the Grassmanns have done in their Ausdehnungslehre series,
In the course of doing this Hermann revisited every fundamental notion in science, really deconstructing Euclid and resynthesising his Stoikeioon. This was Justus stated goal. Hermann got interested in it at a stage when Justus had implemented school trials of many of his ideas and reformulations, picking out an inner structure that Justus could not apprehend because he was the pathfinder! Finally Robert extended the analysis to the foundation of philosophy itself! He called this approach Formenlehre, and really engaged with the Platonic philosophy or theory of forms and Ideas.
That Euclid should serve to unite a constructive approach to geometry, and a empirical approach to philosophy says to me that the Stoikeioon is not a book about geometry or mathematics. It is a book about the philosophy of Form/ ideas.
Hermann literally was between these 2 seminal path finding researchers. He did not live in the same household as his father and brother, but with his uncle. This was a child adoption arrangement, but not an abandonment by any means. Hermann seems to have always known who his biological father was. But he lived with and was brought up in his uncles family due to infertility it seems.
Hermann's absorption of Euclid was thus mostly unconscious and lifelong. It was not as a classical training in Euclid, but as a result of an innovative programme of restructuring geometrical education that he imbibed Euclidean forms. Much the same happened in my education. When geometry was taught , it was not to understand Euclid it was as a subject called Geometry. It was not until much much later in life that I actually began to look at the Greek text!
What that does is dissociate the student from the originator. The student never actually knows what the original thought was. This is why I read Grassmann's own words, in German or Prussian .
The second point is, by the time you do get to study the original the prejudice has already been imposed, so the original thought is almost indiscernible!
The advantage that those who have a classics education have is that through learning the actual language they can " rinse" their brains of prejudicial ideas. The more languages they learn to meditate in, the clearer their conceits become.. But then they face the problem of dissociation: they no longer think the same as those who are brain washed differently!
Any way. Newton read the original Greek text or a Latin copy. Thus the quality of his thinking and insight was suffused and informed by Platonic ideals. Hamilton read Euclid in the Greek and was similarly motivated. The Grassmanns had access to the notions of geometry associated to Euclid. Justus clearly had experience of parts of the text, but I do not know yet which parts or books.
Hermann was a great linguist and that makes me think he had access to the texts either directly or through deconstructed copies in the form of lexicons. Someway, somehow, the ideas and forms of Euclid suffused all the above mentioned individuals.
Thus when Newton established his " vector" algebra based on the parallelogram , it was not in isolation. Hooke, Huygens and all the geometers knew precisely what he meant. But when Grassmann similarly deconstructed and resynthesized the geometry of the parallelogram, suddenly he is talking " obscurely"?
The only difference between Hermann and Newton is the subject boundary called algebra. In newtons day algebra was not a subject, it was included within general Rhetoric and Reasoning. Reason is derive from ratio and proportion, logos and Kairos in Greek, and thus not considered part of the mathematical subject boundary. It was and still is within the Philosophy subject boundary.
Thus we have this academic compartmental approach blocking understanding for millennia! The shake up and mixing of the Humboldt reforms , creating the mix of primary and higher academic teachers , alongside the philosophical argument about the structure and nature of mathematics and the sciences engendered a school of thought that knowledge is constructed, not discoursed! Discoursed means several things all at the same time: but the essential idea is that you have to run round all over the place discovering through inquiry, discussion and debate what may be called divine or spiritual Truths, which you intuitively know to be true!
The constructionists basically say, truth is not the criteria, fact is. It is true because it is fact, that is because it has been constructed!
Thus Justus enters the fray , utilising the growing conception of ring and group theory as expounded in crystallography. He actually makes a fundamental contribution to Verbindungslehre as it was called before it came to be dominated by others who renamed it Kôrperlehre!.
To get to the point , an analytical approach to geometry in order to reconstruct it on a systematic, logical and congruent basis revealed a repeated combinatorial structure in every geometry: the geometry of the line, the geometry of the plane, and in the geometry of the Raume , that is 3d geometry. This structure was deliberately based on Arithmetic, because it was believed arithmetic was logically pure and unsullied.
Justus analysis showed however that this was not the case, and he made several suggestions at how to put it right. Some of which were conceptually confused. He could not resolve certain logical difficulties without bringing in the observer as a crucial part. This actually meant that general rules were subject to the individual assenting to a consensus. In those days they believed that the consensus could get it right , even though they were agitating against an old consensus which they felt was wrong!
So Grassmann J, H and R were looking at the ring or group structure of various geometries and finding connections between them. When Hermann looked at the triangle he clearly picked out an additive Algebra! He was looking at geometry and he could see an Algebra. It was only manifest when the correct Notation was used.
Hmm.. Interesting but not earth shattering, until he noticed the same thing when he was looking at the Geometry of the Quadrilaterals. Again, looking at geometry , with the appropriate Notation revealed an Algebra, but this time it was multiplicative! And in addition it connected with the triangular algebra of addition to produce a distributive rule of combination!
This was so intriguing that he began to explore it and found that the " Algebra" held true. Testing it a bit harder he put in the metrics of length and it still held true. It was when he put in the notion of direction that his world turned upside down! The factors , if you could call them that in the analogy did not commute, but instead required the sign that denoted direction to change when they change position.
After waiting some time to get over the shock and general unease at his conclusions he tested them over and over and found them to be logically consistent. He then decided to devote his life's work to exploring these geometric Algebras. There was much work to be done, many gaps in the algebras to explore, but his hard work and dedication to detail, following the strict guidelines of his father seemed to be being rewarded handsomely. He entered and won mathematical competitions to the astonishment of all around him! He found independent confirmation of his ideas in other researchers work. He read and digested Lagrangian Celestial. Mechanics with astonishing ease because his insight into the geometrical algebra suddenly made it simpler clearer, more symmetrical and beautiful!
In tackling the problem of Ebb and Flow he discovered not only the nascent hyperbolic geometry, but his insight revealed its fundamental algebra. The algebra of Newtonian vectors, as understood by Lagrange was suddenly placed before him, and he could clearly see the parallelogram and in this case the rectangular parallelogram geometric algebra.
Newton and thus Lagrange had fully worked this algebra out. In fact most researchers like Huygens, Leibniz Hooke for example were fully conversant with it. But it was called " algebra" only loosely. It was to Newton his own private cogitation by which he mentally manipulated ideas and relationships to gain insights and find solutions. Although Newton was highly organised and structured in his thinking, he did not see that that was important enough to publish.
Bombelli is probably the first author in this epoch to write a book mostly about Algebra. Descartes is the next author,of renown, but he called it Geometry. De Fermat popularised this geometric algebra, but it was Wallis that, drawing upon Euclid and Barrow wrote the first real modern book on Cartesian Geometric Algebra with his great insight. He pestered Newton for his algebraic notations, because he believed that through studying the algebraic reasonings of genius, students could benefit and emulate and surpass. Thus Wallis's work was the standard for Algebra for a long time. And it was always a geometric algebra.
The Grassmanns were different. They studied the Algebras of the geometries, not the algebra of the geometers!
The algebra of the geometries as I explained above required appropriate notation or terminology to distinguish. Thus Hermann struggled to find for each geometry that appropriate notation that made its appearance manifest, or clear, visible(Anschaung). This was like making ghosts or spirits visible. It was like making subjective notions, ideas and forms visible. It was giving form to an invisible structure of formal thinking , revealing how it followed similar and analogous patterns in all the geometries.
So when Hermann came upon the projective geometries, especially the Newtonian decomposition of " vectors" as forces or velocities, he recognised that it was an algebra that applied like an algorithm to any description of physical situations.. He might have petered out at this stage, because essentially he was going to be repeating Newtons Classic Principia. He would have brought little that was new to the discussion. However, his strong notion of geometric algebra in a given geometry lead him to look for addition, multiplication analogues in the projective geometry. He was able to bypass the detail and see the product, and how the parallelogram formed the general product( product here means the constructed form, which Hermann long ago had convinced himself was an analogue of multiplication). But then he found the inner product and with it division!
In the parallelogram geometry Hermann had been able to see that addition and multiplication algebras existed. He therefore knew that if he could solve for the parallelogram he could solve any problem that could be reduced to its form. However, because there was no sense of division, the algebra of the parallelogram geometry was incomplete. In fact it was full of holes.
Herrmann was already considerably blessed with simplifications due to his earlier discoveries. His reasoning and equation formulation was already considerably shorter and smoother than his contemporaries, simply by using this geometric algebra through appropriate notation as his page layout. What I mean by this is that mathematical notation is set out on the page. If you organise it carefully you can make beneficial use of that layout. To speed up and streamline calculation.
The organisation of the calculation on the page has always been a pedagogical concern. It was clear that " neatness" helped in solving problems and performing calculations accurately.rows and columns have always been a significant part of the mathematical discipline since Babylonian script was invented and tablets of information recorded. But the crucial geometric structure of a page layout is derived not from cuneiform, but from Mosaics or Arithmoi.
An Arithmos is not just a mosaic it is the fundamental of geometry itself! As a fundamental of geometry it is used to record Astrology( astro-metry,astro-nomy astro-logia). Thus these mosaics become fundamental and obscured organising principles. Onto such a mosaic a geometrical form may be drawn, the mosaic then representing an epipedos epiphaneia, a so calle flat surface, but in fact often a very colourful abstract art form we now call a mosaic.
"On such a platform geometry can be done !" mused Pythagoras. And of course he was right. Geometry is always done on an embedded mosaic. Because we have lost sight of this we do not understand Arithmos, Arithmoi, and how forms can be shown to be equal without some other notion like length or area or volume. For Greek mathematics, these fundamental notions are embedded in the mosaic. They are the literal structure of space .
So the layout on the page was also in Hermanns mind and when he discovered the Type he called the inner product he states" but this notation also places another product ( to the exterior product as he soon calls it) TO THE side! ( zur Seite), by this I think he means that alongside the exterior product one must also, on the page work out the interior product. The reason is that when both these are done they are a proportion or ratio( by dividing one into the other a fraction) that has a valid value. This value he realised uniquely replaces the angle in parallelogram geometry and provides proportion,ratio and division into the algebraic tool box!
He knew this to be true because he had done the work in his Ebb and Flow paper In which he had used the hyperbolic sin and cosine to create a single multiple form involving a Strecken and an exponential function.
At the time he had taken it as evidence of the parallelogram multiplicative distribution rule, what I called the law of 3 Strecken. But now his insight into the inner product made him realise it was based on the projected Strecken ( obviously!) and not just any Strecken and the angle between them. The angle was not the important ratio of magnitude . A more general ratio of magnitude was involved and that was that of the outer and inner product!
Angle has long been one of those unquestioned measuring algorithms. But it needed to be questioned, and Newton, Cotes and DeMoivre did question it, particularly as it did not concur with astronomical practice. Astronomers use arcs and always have. Someone, some teacher converted the arc into the angle notion and created a serious problem. Or rather later generations misread the symbol for arc and mistakenly developed the notion of angle. Cotes apparently suggested the radian as an arc measure which could easily be used astronomically and geometrically for mundane land measure. The Greeks used Chords, not angles. Each chord has an associated arc, and the 2 together form the connecting link to orbital motions in the heavens. Thus the straight line measure of spread is precisely what the sine and cosine tables record in ratio form. We give a precise ratio of cord to diameter to measure a circular arc. We may never be able to measure the arc directly but we can construct it and through that solve for the triangle AND for the arc by approximation..
Grassmann had now got a complete ring in his parallelogram Algebra. But more than that he could see how to generalise it to any form made up of facets like a crystal.. This is what he means by n dimensional Algebra! Any complex form made up of parallelograms is n dimensional, depending on how many facets were distinguishable. Each of these facets he called a space, and worked out, purely using his algebraic toolbox for parallelograms how to construct a crystal, and how to distinguish crystal forms.
This is where he found and corrected a mistake made by his father. The fact that he could see it clearly was testament to the powerful tools he had created for describing the Algebra of geometries!
There was yet more research to do, but he wanted to publish his first volume and his results to create a stir and get others involved in the research which was now an extensive and rich field of study. That plan and hope backfired on him disastrously, and almost extinguished his belief in his system. That is why when his brother offered , demanded that he redact and republish his work, insisting that it was not to be left idle!, he was willing to give in to his brother's views of his work. Later, much encouraged by the response he reasserts his own view republishing his unredacted original with annotations.
This is the background to the Grassmanns contribution to a revolution in science.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on May 22, 2013, 02:50:39 PM
That's not quite what I had in mind but thank you very much. :)
I also didn't think you'd jump at this so quickly...
I also didn't think you'd jump at this so quickly...
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Alef on May 22, 2013, 03:59:45 PM
Newtonian triples are in thread in this subsection.
http://www.fractalforums.com/new-theories-and-research/newtonian-triples/ (http://www.fractalforums.com/new-theories-and-research/newtonian-triples/)
Could you give some formula like x=blablabla(x)+c; y=blablabla(y)+c; z=blablabla(z)+c; ? Could be that 4th dimension ruins everything. Quaternion mandelbrots are just boring revolution surfaces. Unitary 4 dimensional equations are revolutions. 4 dimensions probably are very good and easy in mathematics, but 3 dimensional formulas produce more interesting results. Anyway, 4 dimensional space is artificial concept.
What would be formula for 3 dimensional vector mandelbrot?
Maybe 3 variable tensor mandelbrot? It's not just relativity, stress tensors are important in material sciences, throught they must be put wholy into 3 dimensions.
3 dimensional geometric mandelbrot formula?
http://www.fractalforums.com/new-theories-and-research/newtonian-triples/ (http://www.fractalforums.com/new-theories-and-research/newtonian-triples/)
Could you give some formula like x=blablabla(x)+c; y=blablabla(y)+c; z=blablabla(z)+c; ? Could be that 4th dimension ruins everything. Quaternion mandelbrots are just boring revolution surfaces. Unitary 4 dimensional equations are revolutions. 4 dimensions probably are very good and easy in mathematics, but 3 dimensional formulas produce more interesting results. Anyway, 4 dimensional space is artificial concept.
What would be formula for 3 dimensional vector mandelbrot?
Maybe 3 variable tensor mandelbrot? It's not just relativity, stress tensors are important in material sciences, throught they must be put wholy into 3 dimensions.
3 dimensional geometric mandelbrot formula?
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on May 23, 2013, 12:34:31 AM
This is a generic concept but one that is fairly easy to construct. It works for any number of dimensions and you can have three types of dimensions: ones that square to 1, ones that square to -1 and ones that square to 0.
What specific directions square to is called their signature, so if you have a direction a, that squares to 1, a*a=1, that is a positive signature. If a direction b squares to -1, b*b=-1, that's a negative signature, and finally, a direction c squares to 0, c*c, that is a 0 or degenerate signature. A space with those three directions would have signature +,-,0. For most purposes, you'll choose signatures without a 0 but it depends on your applications.
So for instance, for two dimensions x,y, with "signature" -,- , you construct:
A base scalar, name it
1
The two base directions
x
y
and now you combine them by multiplication:
1*1 = 1
1*x = x
1*y = y
x*1 = x
x*x = -1
x*y = xy (this represents a directed unit piece of a plane)
y*1 = y
y*x = -xy (the same unit piece of a plane but "pointing" in the opposite direction)
y*y = -1
ok, so now you have one new object: xy
1*xy = xy
x*xy = (x*x)*y = -1*y = -y
y*xy = -y*yx = -(y*y)*x = x
xy*1 = xy
xy*x = -yx*x = -y*(x*x) = y
xy*y = x*(y*y) = -x
xy*xy = -yx*xy = -y*(x*x)*y = y*y = -1
If you have more directions, you will get more different objects, so for instance, for 3D:
1 (scalar / 0 - vector)
x (1-vector)
y
z
xy - represents a unit xy plane segment (2-vectors)
yz - represents a unit yz plane segment
zx - represents a unit zx plane segment
xyz - represents a unit volume space segment (3-vector)
Geometric interpretation of 1-,2- and 3-vectors: (∧-notation explained below)
(http://upload.wikimedia.org/wikipedia/commons/thumb/c/c3/N-vector.svg/200px-N-vector.svg.png)
The arrows in the surfaces represent the orientation.
x, y and z are what makes a typical vector which, in the Geometric Algebra formalism, is called a 1-vector
1, the scalar, is called a 0-vector
the plane segments are called 2- and the volume segment 3-vectors respectively. For higher dimensional space you'll get more and more types of vectors.
Unlike what you would do in a normal vector algebra, you can add any type of vector together, which also means, adding a scalar to a vector is absolutely allowed. Adding them just keeps them together, precisely like a complex number is the addition of a real and an imaginary number.
That way you get a multivector: a+b x+c y+d z+e xy+f yz+g zx+h xyz
Because you can write:
xy*xyz=x*y*x*y*z = -x*y*y*x*z = x*x*z = -z
and so on, it's very convenient to shorten the highest vector-type, in this case the 3-vector, to just "i", so your elements become:
1, x, y, z, ix, iy, iz, i
This already suggests, that i is something like a complex unit and if you do the math, you can easily show that for this 3D-case, ix, iy and iz are equivalent to what would be called i, j and k in a quaternion.
This way, the multivector above becomes: a+bx+cy+dz+fxi+gyi+ezi+hi (I changed the order to correspond to the new naming)
I used one rule here that I didn't mention previously, namely, that vector components generally anti-commute, so:
x*y = - y*x
Also, I mentioned that you can have directions that square to 1 rather than -1. This has applications, for instance, in Minowski Spacetime which has the following rules:
t*t=1 (time direction)
x*x=y*y=z*z=-1
all other multiplications between them anticommute
Back to the 2D example for simplicity:
We have:
1,x,y,xy
(where x²=-1 and y²=-1)
so a generic number here would be:
a+bx+cy+dxy
and squaring it would be done like so:
(a+bx+cy+dxy)² =
a² + abx + acy + adxy + bxa + (bx)² + bxcy + bxdxy + cya + cybx + (cy)² + cydxy + dxya + dxybx + dxycy + (dxy)² =
a² + abx + acy + adxy + abx - b² + bcxy - bdy + acy - bcxy - c² + cdx + adxy + bdy - cdx - d² =
a²-b²-c²-d² + x (ab+ab+cd-cd) + y (ac-bd+ac+bd)+ xy (ad+bc-bc+ad) =
a²-b²-c²-d² + 2ab x + 2ac y +2ad xy
So the vectorial parts and the bivectorial part are scaled by a and doubled, while the scalar part is reduced in size by the other three.
Now let's see the same thing if you take x and y as squaring to 1:
(a+bx+cy+dxy)² =
a² + abx + acy + adxy + bxa + (bx)² + bxcy + bxdxy + cya + cybx + (cy)² + cydxy + dxya + dxybx + dxycy + (dxy)² =
a² + abx + acy + adxy + bxa + b² + bxcy + bxdxy + cya + cybx + c² + cydxy + dxya + dxybx + dxycy - d² =
a² + b² + c² - d² + 2ab x + 2ac y + 2ad xy
This is the result I previously showed already.
Now this is all nice and such, but why is it actually useful?
It turns out, every part of this has a very clear interpretation and it avoids a ton of problems while significantly shortening your usual notation.
For example, if you look at a cross product, it has a whole lot of problems.
First of all, it is exclusively defined in 3D. To apply it in 2D you need to artificially introduce a third coordinate and in 4D it doesn't work at all anymore and you need to find other ways to do the same thing.
And if that wasn't bad enough, there are certain applications where you actually have to consider, that the cross product doesn't give you your average vector. It gives you a so called "axial vector" which behaves just fine under a normal rotation but if you reflect it, its sign changes.
A usual vector won't change sign under reflection.
Now, in Geometric Algebra, you can take the product x*y = xy which gives you a directed piece of a plane. This holds the same information as a cross product but has the additional benefits of working always (in any number of dimensions) and not being broken under reflections.
I previously mentioned that there are three types of product in this formalism.
The product I used thus far is the so called vector product and it simply is worked out like you'd work out the product of a complex number or quaternion, as entirely defined by how to multiply the directional components of a vector.
The other two products, called the wedge and inner product respectively, are defined by the vector product but to show them off, I'll need two different generic vectors. For simplicity, I'll stick to two dimensions (and the components square to one) :
v1 v2 = (ax+by)(cx+dy) = ax cx + ax dy + by cx + by dy =
ac + bd + (ad-bc) xy
v2 v1 = (cx+dy)(ax+by) = cx ax + cx by + dy ax + dy by =
ac + bd + (bc-ad) xy
Note that, if you multiply two pure 1-vectors (what you are used to as being vectors), you'll end up with only a scalar (0-vector) and a bi-vector (2-vector) part and no usual 1-vectors at all.
Now from this, the inner product is defined as:
v1.v2 = v2.v1 = (v1v2 + v2v1)/2 = (ac + bd + (ad-bc) xy + ac + bd + (bc-ad) xy)/2 =
(2ac+2bd + xy (ad-ad+bc-bc))/2 = ac+bd
so the inner product of a 1-vector is a scalar. You may recognize this as the dot-product. It gives the cosine of the angle between the two vectors times their lengths and in case the two vectors are normal on each other, it will be zero.
Geometric interpretation of the inner product:
(http://upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Dot-product-2.svg/200px-Dot-product-2.svg.png)
And the wedge-product is defined as:
v1∧v2 = -v2∧v1 = (v1v2 - v2v1)/2 = (ac + bd + (ad-bc) xy - (ac + bd + (bc-ad) xy))/2 =
(ac-ac+bd-bd+xy (ad+ad-bc-bc))/2 = (ad-bc) xy
so the wedge product of a 1-vector is a bivector. There is no exact equivalent to this in normal vector-algebra, but if you compare it to the cross-product, it is very similar to that and as such it gives the signed area of the parallelogram spanned by the two input vectors. Thus it is like a normal vector, except that it doesn't behave oddly under reflections and you can do this very same thing in any number of dimensions without running into problems.
Comparison (only possible in 3D) between the Wedge product and the Cross product:
(http://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Exterior_calc_cross_product.svg/200px-Exterior_calc_cross_product.svg.png)
So the inner and the wedge product are just the symmetric and anti-symmetric parts of the full vector product. But unlike the scalar or cross product which are exclusively defined for (1-)vectors, you can take the same definitions and apply them to multi-vectors. And if you delve a bit into the matter, it turns out that these extended definitions also have very real geometric interpretations. - This is one of the main strengths of this formalism: It allows incredible amounts of abstractions and yet everything has a clear, tangible geometric interpretation that you could even intuitively draw in 2D or 3D if you want.
Now, the precise relationship between the wedge product in 3D and the cross product is:
a x b = i a ∧ b, where i, in this case, would be xyz and a and b are (1-)vectors.
By simple multiplication, you can try for yourself and conclude, that using i like this turns a bivector (2-vector; which results from a ∧ b) into a vector (which is what you expect to get from a x b).
TO BE CONTINUED (in another post)
What specific directions square to is called their signature, so if you have a direction a, that squares to 1, a*a=1, that is a positive signature. If a direction b squares to -1, b*b=-1, that's a negative signature, and finally, a direction c squares to 0, c*c, that is a 0 or degenerate signature. A space with those three directions would have signature +,-,0. For most purposes, you'll choose signatures without a 0 but it depends on your applications.
So for instance, for two dimensions x,y, with "signature" -,- , you construct:
A base scalar, name it
1
The two base directions
x
y
and now you combine them by multiplication:
1*1 = 1
1*x = x
1*y = y
x*1 = x
x*x = -1
x*y = xy (this represents a directed unit piece of a plane)
y*1 = y
y*x = -xy (the same unit piece of a plane but "pointing" in the opposite direction)
y*y = -1
ok, so now you have one new object: xy
1*xy = xy
x*xy = (x*x)*y = -1*y = -y
y*xy = -y*yx = -(y*y)*x = x
xy*1 = xy
xy*x = -yx*x = -y*(x*x) = y
xy*y = x*(y*y) = -x
xy*xy = -yx*xy = -y*(x*x)*y = y*y = -1
If you have more directions, you will get more different objects, so for instance, for 3D:
1 (scalar / 0 - vector)
x (1-vector)
y
z
xy - represents a unit xy plane segment (2-vectors)
yz - represents a unit yz plane segment
zx - represents a unit zx plane segment
xyz - represents a unit volume space segment (3-vector)
Geometric interpretation of 1-,2- and 3-vectors: (∧-notation explained below)
(http://upload.wikimedia.org/wikipedia/commons/thumb/c/c3/N-vector.svg/200px-N-vector.svg.png)
The arrows in the surfaces represent the orientation.
x, y and z are what makes a typical vector which, in the Geometric Algebra formalism, is called a 1-vector
1, the scalar, is called a 0-vector
the plane segments are called 2- and the volume segment 3-vectors respectively. For higher dimensional space you'll get more and more types of vectors.
Unlike what you would do in a normal vector algebra, you can add any type of vector together, which also means, adding a scalar to a vector is absolutely allowed. Adding them just keeps them together, precisely like a complex number is the addition of a real and an imaginary number.
That way you get a multivector: a+b x+c y+d z+e xy+f yz+g zx+h xyz
Because you can write:
xy*xyz=x*y*x*y*z = -x*y*y*x*z = x*x*z = -z
and so on, it's very convenient to shorten the highest vector-type, in this case the 3-vector, to just "i", so your elements become:
1, x, y, z, ix, iy, iz, i
This already suggests, that i is something like a complex unit and if you do the math, you can easily show that for this 3D-case, ix, iy and iz are equivalent to what would be called i, j and k in a quaternion.
This way, the multivector above becomes: a+bx+cy+dz+fxi+gyi+ezi+hi (I changed the order to correspond to the new naming)
I used one rule here that I didn't mention previously, namely, that vector components generally anti-commute, so:
x*y = - y*x
Also, I mentioned that you can have directions that square to 1 rather than -1. This has applications, for instance, in Minowski Spacetime which has the following rules:
t*t=1 (time direction)
x*x=y*y=z*z=-1
all other multiplications between them anticommute
Back to the 2D example for simplicity:
We have:
1,x,y,xy
(where x²=-1 and y²=-1)
so a generic number here would be:
a+bx+cy+dxy
and squaring it would be done like so:
(a+bx+cy+dxy)² =
a² + abx + acy + adxy + bxa + (bx)² + bxcy + bxdxy + cya + cybx + (cy)² + cydxy + dxya + dxybx + dxycy + (dxy)² =
a² + abx + acy + adxy + abx - b² + bcxy - bdy + acy - bcxy - c² + cdx + adxy + bdy - cdx - d² =
a²-b²-c²-d² + x (ab+ab+cd-cd) + y (ac-bd+ac+bd)+ xy (ad+bc-bc+ad) =
a²-b²-c²-d² + 2ab x + 2ac y +2ad xy
So the vectorial parts and the bivectorial part are scaled by a and doubled, while the scalar part is reduced in size by the other three.
Now let's see the same thing if you take x and y as squaring to 1:
(a+bx+cy+dxy)² =
a² + abx + acy + adxy + bxa + (bx)² + bxcy + bxdxy + cya + cybx + (cy)² + cydxy + dxya + dxybx + dxycy + (dxy)² =
a² + abx + acy + adxy + bxa + b² + bxcy + bxdxy + cya + cybx + c² + cydxy + dxya + dxybx + dxycy - d² =
a² + b² + c² - d² + 2ab x + 2ac y + 2ad xy
This is the result I previously showed already.
Now this is all nice and such, but why is it actually useful?
It turns out, every part of this has a very clear interpretation and it avoids a ton of problems while significantly shortening your usual notation.
For example, if you look at a cross product, it has a whole lot of problems.
First of all, it is exclusively defined in 3D. To apply it in 2D you need to artificially introduce a third coordinate and in 4D it doesn't work at all anymore and you need to find other ways to do the same thing.
And if that wasn't bad enough, there are certain applications where you actually have to consider, that the cross product doesn't give you your average vector. It gives you a so called "axial vector" which behaves just fine under a normal rotation but if you reflect it, its sign changes.
A usual vector won't change sign under reflection.
Now, in Geometric Algebra, you can take the product x*y = xy which gives you a directed piece of a plane. This holds the same information as a cross product but has the additional benefits of working always (in any number of dimensions) and not being broken under reflections.
I previously mentioned that there are three types of product in this formalism.
The product I used thus far is the so called vector product and it simply is worked out like you'd work out the product of a complex number or quaternion, as entirely defined by how to multiply the directional components of a vector.
The other two products, called the wedge and inner product respectively, are defined by the vector product but to show them off, I'll need two different generic vectors. For simplicity, I'll stick to two dimensions (and the components square to one) :
v1 v2 = (ax+by)(cx+dy) = ax cx + ax dy + by cx + by dy =
ac + bd + (ad-bc) xy
v2 v1 = (cx+dy)(ax+by) = cx ax + cx by + dy ax + dy by =
ac + bd + (bc-ad) xy
Note that, if you multiply two pure 1-vectors (what you are used to as being vectors), you'll end up with only a scalar (0-vector) and a bi-vector (2-vector) part and no usual 1-vectors at all.
Now from this, the inner product is defined as:
v1.v2 = v2.v1 = (v1v2 + v2v1)/2 = (ac + bd + (ad-bc) xy + ac + bd + (bc-ad) xy)/2 =
(2ac+2bd + xy (ad-ad+bc-bc))/2 = ac+bd
so the inner product of a 1-vector is a scalar. You may recognize this as the dot-product. It gives the cosine of the angle between the two vectors times their lengths and in case the two vectors are normal on each other, it will be zero.
Geometric interpretation of the inner product:
(http://upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Dot-product-2.svg/200px-Dot-product-2.svg.png)
And the wedge-product is defined as:
v1∧v2 = -v2∧v1 = (v1v2 - v2v1)/2 = (ac + bd + (ad-bc) xy - (ac + bd + (bc-ad) xy))/2 =
(ac-ac+bd-bd+xy (ad+ad-bc-bc))/2 = (ad-bc) xy
so the wedge product of a 1-vector is a bivector. There is no exact equivalent to this in normal vector-algebra, but if you compare it to the cross-product, it is very similar to that and as such it gives the signed area of the parallelogram spanned by the two input vectors. Thus it is like a normal vector, except that it doesn't behave oddly under reflections and you can do this very same thing in any number of dimensions without running into problems.
Comparison (only possible in 3D) between the Wedge product and the Cross product:
(http://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Exterior_calc_cross_product.svg/200px-Exterior_calc_cross_product.svg.png)
So the inner and the wedge product are just the symmetric and anti-symmetric parts of the full vector product. But unlike the scalar or cross product which are exclusively defined for (1-)vectors, you can take the same definitions and apply them to multi-vectors. And if you delve a bit into the matter, it turns out that these extended definitions also have very real geometric interpretations. - This is one of the main strengths of this formalism: It allows incredible amounts of abstractions and yet everything has a clear, tangible geometric interpretation that you could even intuitively draw in 2D or 3D if you want.
Now, the precise relationship between the wedge product in 3D and the cross product is:
a x b = i a ∧ b, where i, in this case, would be xyz and a and b are (1-)vectors.
By simple multiplication, you can try for yourself and conclude, that using i like this turns a bivector (2-vector; which results from a ∧ b) into a vector (which is what you expect to get from a x b).
TO BE CONTINUED (in another post)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on May 23, 2013, 01:54:09 AM
Alef, all this is an extension / unification of vectors with complex numbers or quaternions, so from most naive formulae (that is, just directly squaring the numbers and straight on adding a constant), you'll likely always get a revolution surface. However, I think it already was show before by someone, that all the triplex algebras can be fully encoded in quaternion algebras. This, if I recall correctly, was part of the process of getting the initially trigonometric formulae into a trig-free form that would be faster to compute.
If you want interesting structures from these, you can either try a direct geometric approach (thinking of how rotations, reflections and scalings actually cause dynamic, interesting strange attractors in the first place) or you can just mess around with the definitions so you have something less symmetric (which thus would have a more complicated description in quaternions or geometric algebra)
The interesting part of all this is the wide applicability and strong geometric connection, so you can, for example, rather straight-forwardly come up with interesting experiments based on the direct geometry of fractal dynamics, rather than more or less blindly messing around with numbers.
Recent posts showed me that, for example, pauldelbrot has this geometric intuition very much internalized (http://www.fractalforums.com/new-theories-and-research/spacefilling-julia-and-mandelbrot-fractals/msg61447/#msg61447) at least for the 2D case. This is a way of giving the same level of insight to any dimension. (Though the particular idea, Pauldelbrot had in that post, relates to projective space which will probably need some extra coverage)
If you want interesting structures from these, you can either try a direct geometric approach (thinking of how rotations, reflections and scalings actually cause dynamic, interesting strange attractors in the first place) or you can just mess around with the definitions so you have something less symmetric (which thus would have a more complicated description in quaternions or geometric algebra)
The interesting part of all this is the wide applicability and strong geometric connection, so you can, for example, rather straight-forwardly come up with interesting experiments based on the direct geometry of fractal dynamics, rather than more or less blindly messing around with numbers.
Recent posts showed me that, for example, pauldelbrot has this geometric intuition very much internalized (http://www.fractalforums.com/new-theories-and-research/spacefilling-julia-and-mandelbrot-fractals/msg61447/#msg61447) at least for the 2D case. This is a way of giving the same level of insight to any dimension. (Though the particular idea, Pauldelbrot had in that post, relates to projective space which will probably need some extra coverage)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: jehovajah on May 25, 2013, 05:55:16 PM
Some further research Background for those interested.
http://my.opera.com/jehovajah/blog/2013/05/25/the-gnomonic-algebra-of-the-stoikeioon-book-2
http://my.opera.com/jehovajah/blog/2013/05/25/the-gnomonic-algebra-of-the-stoikeioon-book-2
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on June 01, 2013, 12:47:11 AM
This might be accessible to more people if you describe how GA is related to stuff they might already know. For example, in terms of Clifford/GA notions then a complex number is a scalar plus a bivector. Oh and a quaternion is a scalar plus a bivector. (even sub-algebras: grade-0 and grade-2) The bivector in 2D seems likes a scalar and in 3D like a vector because they are represented by 1 & 3 components respectively, but in both cases they're oriented 2D subspaces.
If my brain is working correctly then the standard 2D model of GA is the same as a dual-number of complex-numbers and 3D is a dual number of quaternions.
If my brain is working correctly then the standard 2D model of GA is the same as a dual-number of complex-numbers and 3D is a dual number of quaternions.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on June 01, 2013, 11:32:19 PM
pretty much, yes.
I only started looking into all this not too long ago. While I already have a rough, intuitive grasp of it all (the straight forward geometric interpretation makes this remarkably easy), I haven't fully internalized this yet, so I can't really write it up nicely.
However, note that I put a couple of nice resources that already did so in the first post of this thread.
I only started looking into all this not too long ago. While I already have a rough, intuitive grasp of it all (the straight forward geometric interpretation makes this remarkably easy), I haven't fully internalized this yet, so I can't really write it up nicely.
However, note that I put a couple of nice resources that already did so in the first post of this thread.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on June 04, 2013, 07:45:52 PM
Ok, so I manually computed the multiplication matrix for the so called space-time-algebra which corresponds to the minkowski space. Note that this is quite some work to do manually and it's rather error prone. I do hope that it's fully correct but it could be that I accidentally flipped a sign here or there. If you notice an error, please point it out.
There you have it:
(http://puu.sh/38cte.png)
The colors all have a meaning. They essentially tell you, which directions are included where.
I chose to assign xyz to RGB and, since there isn't a fourth color, t to be a dark grey.
"1" - the scalar - is a bright grey, while "i" - the pseudo scalar, is black.
The planes and 3-planes are color-coded accordingly. So for instance, the xy-plane is red and green, e.g. yellow, from x (red) and y (green).
Furthermore, the white lines split it into conceptual differences. Space is grouped together, time, the scalars and the pseudo-scalars are split off.
If you look through what each element does, you'll hopefully see the geometric meaning. So for instance:
x*xy = -y
y*xy= x
So if you multiply a vector that lies within the xy plane with that plane, you'll rotate that vector by 90°. (If you have a right-handed coordinate system, it will rotate counterclockwise)
"i" is sort of a negation operator. For instance, i*x=ix = txyzx = -tyz
So i*x means "all vectors except for x", or put differently, "all vectors, normal to x"
i times a plane will give you the plane that is perpendicular to the current plane.
etc.
The special thing about minkowski space is, that time commutes while space anti-commutes.
So, t²=1, while x², y² and z² all = -1
The space thus is different from sedenions which you would get if t² = -1 too.
Every plane will act on vectors inside that plane like the "i" of complex numbers would act in the complex plane.
Every space will act on vectors inside that space by giving you the plane that is normal to the given vector.
It will act on planes inside the space by giving you the vectors normal to the plane.
And "i" turns every element into what ever is normal to that element.
Vector lengths can be found by just taking the vectorial parts:
(a t + b x + c y + d z)(a t + b x + c y + d z) = a² t² + b² x² + c² y² + d² z² = a²-b²-c²-d²
This corresponds to how distances work in relativistic physics:
Everything always moves at the speed of light. Though, if it doesn't seem to move through space at that speed, it instead will move through time at maximum speed. If you speed up in space, you have to slow down in time, so you'll get a constant speed of light. In other words, the faster you go, the slower time goes for you.
Note though, that this, in length-terms, means, that you mathematically can get vectors of negative length (hyper-relativistic) or even zero length (when distances in space and in time are "equal", this is the light cone, e.g. it describes how photons would travel in a flat spacetime)
Just play around with this for a bit. Also experiment with which elements you actually try to change. For instance, see what happens if you only initialize non-zero values over the six planes, or the four directions, or only the three spacial directions... Plenty of stuff could be done with this.
There you have it:
(http://puu.sh/38cte.png)
The colors all have a meaning. They essentially tell you, which directions are included where.
I chose to assign xyz to RGB and, since there isn't a fourth color, t to be a dark grey.
"1" - the scalar - is a bright grey, while "i" - the pseudo scalar, is black.
The planes and 3-planes are color-coded accordingly. So for instance, the xy-plane is red and green, e.g. yellow, from x (red) and y (green).
Furthermore, the white lines split it into conceptual differences. Space is grouped together, time, the scalars and the pseudo-scalars are split off.
If you look through what each element does, you'll hopefully see the geometric meaning. So for instance:
x*xy = -y
y*xy= x
So if you multiply a vector that lies within the xy plane with that plane, you'll rotate that vector by 90°. (If you have a right-handed coordinate system, it will rotate counterclockwise)
"i" is sort of a negation operator. For instance, i*x=ix = txyzx = -tyz
So i*x means "all vectors except for x", or put differently, "all vectors, normal to x"
i times a plane will give you the plane that is perpendicular to the current plane.
etc.
The special thing about minkowski space is, that time commutes while space anti-commutes.
So, t²=1, while x², y² and z² all = -1
The space thus is different from sedenions which you would get if t² = -1 too.
Every plane will act on vectors inside that plane like the "i" of complex numbers would act in the complex plane.
Every space will act on vectors inside that space by giving you the plane that is normal to the given vector.
It will act on planes inside the space by giving you the vectors normal to the plane.
And "i" turns every element into what ever is normal to that element.
Vector lengths can be found by just taking the vectorial parts:
(a t + b x + c y + d z)(a t + b x + c y + d z) = a² t² + b² x² + c² y² + d² z² = a²-b²-c²-d²
This corresponds to how distances work in relativistic physics:
Everything always moves at the speed of light. Though, if it doesn't seem to move through space at that speed, it instead will move through time at maximum speed. If you speed up in space, you have to slow down in time, so you'll get a constant speed of light. In other words, the faster you go, the slower time goes for you.
Note though, that this, in length-terms, means, that you mathematically can get vectors of negative length (hyper-relativistic) or even zero length (when distances in space and in time are "equal", this is the light cone, e.g. it describes how photons would travel in a flat spacetime)
Just play around with this for a bit. Also experiment with which elements you actually try to change. For instance, see what happens if you only initialize non-zero values over the six planes, or the four directions, or only the three spacial directions... Plenty of stuff could be done with this.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: jehovajah on June 06, 2013, 05:43:25 AM
Nice!
Believe you me when i say i know how time consuming this is to do. I like the colour legend too.
I am still working on V9 and V18!
Believe you me when i say i know how time consuming this is to do. I like the colour legend too.
I am still working on V9 and V18!
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on June 06, 2013, 01:37:53 PM
Thank you.
Yeah, the color legend is nice though I feel like it might be a bit much. Like, this color scheme gives you all information, save for the sign of an outcome.
But it hides lower level information which now is harder to see. I tried to fix that with the white grid-lines, to give an additional information grouping layer and guide the eye a bit more, but it's not as obvious as I'd like it to be.
I'd love to see what patterns are in this that are currently hidden by the presentation. Some I saw just from hand-computing this table...
Yeah, the color legend is nice though I feel like it might be a bit much. Like, this color scheme gives you all information, save for the sign of an outcome.
But it hides lower level information which now is harder to see. I tried to fix that with the white grid-lines, to give an additional information grouping layer and guide the eye a bit more, but it's not as obvious as I'd like it to be.
I'd love to see what patterns are in this that are currently hidden by the presentation. Some I saw just from hand-computing this table...
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kloniwotski on June 20, 2013, 01:40:41 PM
Hi, I've wondered for a long time about the relevance of GA to extending fractals like the Mandelbrot into 3 dimensions (since I first read about the mandelbulb on this site).
Having little mathematical skill, I haven't been able to post anything before. But today I noticed a thesis:
http://www2.eng.cam.ac.uk/~rjw57/pdf/r_wareham_pdh_thesis.pdf
that proposes a simple extension of the operation of squaring a complex number to "squaring" a vector in n dimensions courtesy of the power of GA. (It also gives a pretty good introduction to GA, as far as I can tell.)
The extension is (I think) : v goes to ve1v (using the geometric product) for some basis vector e1 (though I'd encourage anyone interested to look at the pdf rather than go off my probably incorrect relaying of it!)
pseudocode is provided to draw the generalised Mandelbrot:
(The expression "re1r + c" presumes some implementation of the geometric product, I expect)
From looking at the illustration given in the pdf, this allows calculation of a slice through a 3D mandelbrot, presumably depending on the set of "image points". In the illustrations, 3 orthogonal planes are shown. It looks like a 3D structure *might* be able to be rendered with a voxel approach.
I've tried to search for any reference to this thesis on the fractalforums site, but came up with nothing, so I wondered if more mathematically astute members might comment on the relevance or otherwise to 3D mandelbrot extensions? I'd be very interested in this as it's something I've been wondering about (occasionally) for years!
(While searching, though, I did find this page http://www.physicsforums.com/showthread.php?t=206844 where twinbee is recommended to look at a paper whose authors include the writer of the thesis, while still working towards the original mandelbulb, it seems!)
("sub" button isn't working for me in preview, btw. "e1" is meant to be e with a subscript 1, if it doesn't appear that way.)
Having little mathematical skill, I haven't been able to post anything before. But today I noticed a thesis:
http://www2.eng.cam.ac.uk/~rjw57/pdf/r_wareham_pdh_thesis.pdf
that proposes a simple extension of the operation of squaring a complex number to "squaring" a vector in n dimensions courtesy of the power of GA. (It also gives a pretty good introduction to GA, as far as I can tell.)
The extension is (I think) : v goes to ve1v (using the geometric product) for some basis vector e1 (though I'd encourage anyone interested to look at the pdf rather than go off my probably incorrect relaying of it!)
pseudocode is provided to draw the generalised Mandelbrot:
Code:
Require: Set I of vectors associated with image points
imax := maximum number of iterations
e1 := a unit vector in some preferred direction
for all c in I do
r := c
i := 0
while (r * r) < 4 and i < imax do
r := re1r + c
i := i + 1
end while
set pixel c to colour i
end for
(The expression "re1r + c" presumes some implementation of the geometric product, I expect)
From looking at the illustration given in the pdf, this allows calculation of a slice through a 3D mandelbrot, presumably depending on the set of "image points". In the illustrations, 3 orthogonal planes are shown. It looks like a 3D structure *might* be able to be rendered with a voxel approach.
I've tried to search for any reference to this thesis on the fractalforums site, but came up with nothing, so I wondered if more mathematically astute members might comment on the relevance or otherwise to 3D mandelbrot extensions? I'd be very interested in this as it's something I've been wondering about (occasionally) for years!
(While searching, though, I did find this page http://www.physicsforums.com/showthread.php?t=206844 where twinbee is recommended to look at a paper whose authors include the writer of the thesis, while still working towards the original mandelbulb, it seems!)
("sub" button isn't working for me in preview, btw. "e1" is meant to be e with a subscript 1, if it doesn't appear that way.)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on June 20, 2013, 01:56:15 PM
I don't have the time right now to look through this but as far as a quick glance told me, it looks amazing. Thanks for sharing that!
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on June 20, 2013, 03:06:05 PM
The squaring part is easy as I'm poorly explaining here: http://www.fractalforums.com/new-theories-and-research/do-quaternions-need-revisiting/. From a quick skim it looks like the author is missing the geometric connection and could be using a rotor in GA. The only reason I can see to jump to conformal GA would be to perform folds on some conformal mapping or to build a fractal from the various higher order supported primitives.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kloniwotski on June 20, 2013, 03:37:36 PM
Hi Roquen, I'm guessing you're referring to
about the squaring. Unfortunately (did I mention I'm a math noob?) I don't know what the * in this is. "p" and "q" are quaternions, right?
I think Mr Wareham is doing something a bit different to this, not involving rotors, but as I may have mentioned, I'm a bit out of my depth! :)
From what I understand, he shows a mapping from vectors in R2 to numbers in the complex plane r => C(r), and then from that shows that C(re1r) = (C(r))2.
Since the dimensionality / coordinates don't enter into his mapping, the restriction that r is in R2 is not required, so that the re1r operation should be "geometrically analogous" to squaring a complex number for any dimensionality of r. I'm not sure, but it seems the conformal apparatus isn't necessary for the section of the thesis to do with extending the Mandelbrot.
I think the important thing (which I think I stated badly) is that this isn't the same as "squaring a vector in Rn", but is rather "doing something with a vector in Rn that is like squaring a complex number", since it is the special things that happen when squaring complex numbers that produce interesting results in the m-set.
Now for all I know, he may have just faked his picture of the 3D Mandelbrot and his formula might be piffle, but I don't think it's quaternionic anyhow :)
Again I feel the need to plead my lack of mathematical ability - I could well be wrong, but that's how I understood it.
Thanks for the link.
Quote
qpq* where q = sqrt(pr*)
about the squaring. Unfortunately (did I mention I'm a math noob?) I don't know what the * in this is. "p" and "q" are quaternions, right?
I think Mr Wareham is doing something a bit different to this, not involving rotors, but as I may have mentioned, I'm a bit out of my depth! :)
From what I understand, he shows a mapping from vectors in R2 to numbers in the complex plane r => C(r), and then from that shows that C(re1r) = (C(r))2.
Since the dimensionality / coordinates don't enter into his mapping, the restriction that r is in R2 is not required, so that the re1r operation should be "geometrically analogous" to squaring a complex number for any dimensionality of r. I'm not sure, but it seems the conformal apparatus isn't necessary for the section of the thesis to do with extending the Mandelbrot.
I think the important thing (which I think I stated badly) is that this isn't the same as "squaring a vector in Rn", but is rather "doing something with a vector in Rn that is like squaring a complex number", since it is the special things that happen when squaring complex numbers that produce interesting results in the m-set.
Now for all I know, he may have just faked his picture of the 3D Mandelbrot and his formula might be piffle, but I don't think it's quaternionic anyhow :)
Again I feel the need to plead my lack of mathematical ability - I could well be wrong, but that's how I understood it.
Thanks for the link.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on June 20, 2013, 04:13:00 PM
Every person on the planet is a math n00b. Looking at Fig 6.9 the method certainly appears to be equivalent to performing a proper rotation. It also not succeeding in breaking out of the plane.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kloniwotski on June 20, 2013, 04:40:57 PM
Looking at Fig 6.9
6.10, right?
Quote
the method certainly appears to be equivalent to performing a proper rotation. It also not succeeding in breaking out of the plane.
Well, a rotation plus a dilation, it seems. (I hadn't got that far before I posted, in my excitement :) ) The diagram is in 2D, but it looks like for z in R3 it might be the plane in which both e1 and z lie, which of course wouldn't be the same plane for all z.
Edit: Um, in that case it would just be a boring rotated Mandelbrot, would it? Ah well, I had such hopes... :-/
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Alef on June 29, 2013, 07:45:44 PM
Tried to do a fractal using vector cross product. I was looking into old book about math of 1967 so then I don't need to burden myself with cross product being bivector.
If creating another vector Z'=Z+Seed(x,y,z) or Z`=Z+C(x,y,z) cross product would be meaningfull.
So cross product of A1(X1,Y1,Z1) and A2(X2,Y2,Z2).
| Y1 Z1 | Z1 X1| X1 Y1|
A1xA2=( | | , | , | )
| Y2 Z2 | Z2 X2| X2 Y2|
So
X3= Y1xZ2 - Y2xZ1
Y3= X2xZ1 - X1xZ2
Z3= X1xY2 - X2-Y1
Then fractal formula of Z = ZxSeed +Pixel would be:
Zx=real(z); Zy=imag(z); Zz=imag(z);
Cx=real(pixel); Cy=imag(pixel); Cz=imag(pixel);
Sx=real(seed); Sy=imag(seed); Sz=imag(seed);
Z2x= Zy*Sz - Sy*Zz + Cx;
Z2y= Sx*Zz - Zx*Sz + Cy;
Z2z= Zx*Sy - Sx*Zy + Cz;
z=quaternion (Z2x, Z2y, Z2z);
But result was simple as formula. In 3D just something like absolutely smooth stick and in 2D something not great. Don't have pics becouse there were nothing, but realy then it's more simple than quadratic in complex numbers.
If creating another vector Z'=Z+Seed(x,y,z) or Z`=Z+C(x,y,z) cross product would be meaningfull.
So cross product of A1(X1,Y1,Z1) and A2(X2,Y2,Z2).
| Y1 Z1 | Z1 X1| X1 Y1|
A1xA2=( | | , | , | )
| Y2 Z2 | Z2 X2| X2 Y2|
So
X3= Y1xZ2 - Y2xZ1
Y3= X2xZ1 - X1xZ2
Z3= X1xY2 - X2-Y1
Then fractal formula of Z = ZxSeed +Pixel would be:
Zx=real(z); Zy=imag(z); Zz=imag(z);
Cx=real(pixel); Cy=imag(pixel); Cz=imag(pixel);
Sx=real(seed); Sy=imag(seed); Sz=imag(seed);
Z2x= Zy*Sz - Sy*Zz + Cx;
Z2y= Sx*Zz - Zx*Sz + Cy;
Z2z= Zx*Sy - Sx*Zy + Cz;
z=quaternion (Z2x, Z2y, Z2z);
But result was simple as formula. In 3D just something like absolutely smooth stick and in 2D something not great. Don't have pics becouse there were nothing, but realy then it's more simple than quadratic in complex numbers.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Fracturbator on June 30, 2013, 03:05:39 AM
This is a very interesting thread, people - I really want to set aside some time to check all this out when my day-job settles down a bit.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on July 09, 2013, 05:51:27 PM
Ok, so given a vector r=a x + b y + c z and a unit-basis-vecor e, say e=y
The algorithm given in the paper very straight-forwardly is:
r² := r e r =>
(a x + b y + c z) y (a x + b y + c z) =
(a xy + b + c zy) (a x + b y + c z) =
-a² y + ab x + ac xyz + ab x + b² y + bc z - ac xyz + bc z - c² y =
2ab x - (a²-b²+c²) y + 2bc z + 0 xyz
=
)
And from there, you just go the classical route r²+s or z²+c or what ever you like.
So:
iterate:
a=2ab+l
b=-a2+b2-c2+m
c=2bc+n
for a2+b2+c2<4
I'm almost certain this already was tried, but unless I got some sign-error, that's what you get for normal euclidean space.
The fact, that this is symmetric in a and c probably means, this will also be circularly symmetric, so it will likely be another revolution set.
One cool thing the paper does, though, is also allowing for hyperbolic MSets. I'll try to extract that next. It's going to be a bit more complicated though, because it depends on a more complicated expression for distance, using hyperbolic instead of euclidean distance.
The algorithm given in the paper very straight-forwardly is:
r² := r e r =>
(a x + b y + c z) y (a x + b y + c z) =
(a xy + b + c zy) (a x + b y + c z) =
-a² y + ab x + ac xyz + ab x + b² y + bc z - ac xyz + bc z - c² y =
2ab x - (a²-b²+c²) y + 2bc z + 0 xyz
=
And from there, you just go the classical route r²+s or z²+c or what ever you like.
So:
iterate:
a=2ab+l
b=-a2+b2-c2+m
c=2bc+n
for a2+b2+c2<4
I'm almost certain this already was tried, but unless I got some sign-error, that's what you get for normal euclidean space.
The fact, that this is symmetric in a and c probably means, this will also be circularly symmetric, so it will likely be another revolution set.
One cool thing the paper does, though, is also allowing for hyperbolic MSets. I'll try to extract that next. It's going to be a bit more complicated though, because it depends on a more complicated expression for distance, using hyperbolic instead of euclidean distance.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on July 09, 2013, 08:07:45 PM
Like I thought: this yields an identical result to using quaternions. The "problem" is that the fold needs to happen in more than a single plane to do anything interesting. Does anyone have a pointer to an easy to use brute force illum framework I could use?
Indeed one advantage of GA is to change the signature to non-euclidean.
(edit: grammar mistake)
Indeed one advantage of GA is to change the signature to non-euclidean.
(edit: grammar mistake)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on July 09, 2013, 08:19:03 PM
Yeah, I'm not actually surprised about this result. It's kind of a property of rotations to always be in a single plane. You'd need rotations in more than one direction, which brings you to what typically would be a MBulb but ceases to be conformal.
I'm still also thinking about that idea on having a non-conformal transformation that causes infinite smaller details which, however, are not the same as the bigger ones but rather details that keep changing.
So somehow something that controlls the inevidable stretching in a way that manipulates it into aesthetically different structures.
Zooming into that wouldn't give more and more variations of the same but rather just more and more variations.
The hyperbolic MSet in 2 Dimensions looks like this, btw:
(http://puu.sh/3yNDr.jpg)
The border of the circle is actually at infinity, which is why features seem to shrink towards the edges and to be bloated in the middle. However, if you zoom in, this distortion should become less and less notable.
And due to it being different in hyperbolic space to first go straight and then go right vs. first go right and then straight (you'll end up at different points), you actually have two kinds of Julia-sets for the price of a single M-set.
(All the details can be found in the paper)
I'm still also thinking about that idea on having a non-conformal transformation that causes infinite smaller details which, however, are not the same as the bigger ones but rather details that keep changing.
So somehow something that controlls the inevidable stretching in a way that manipulates it into aesthetically different structures.
Zooming into that wouldn't give more and more variations of the same but rather just more and more variations.
The hyperbolic MSet in 2 Dimensions looks like this, btw:
(http://puu.sh/3yNDr.jpg)
The border of the circle is actually at infinity, which is why features seem to shrink towards the edges and to be bloated in the middle. However, if you zoom in, this distortion should become less and less notable.
And due to it being different in hyperbolic space to first go straight and then go right vs. first go right and then straight (you'll end up at different points), you actually have two kinds of Julia-sets for the price of a single M-set.
(All the details can be found in the paper)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on July 11, 2013, 02:50:38 AM
Here's another experiment to try, if that hasn't been done yet:
What if you vary the rotational direction each time?
This is an example where, what would be the "real" axis alternates between the x-axis and the y-axis. Also, you're effectively doing two iterations with each iteration, because that's how I arrived at this closed form.
I differentiate the directions (x,y,z) from the coordinates (a,b,c) and the constant that is added, is denoted by (l,m,n):
x: a -> 2(a²-b²-c²+l)(2ab+m)+l
y: b -> (2ab+m)²-(2ac+n)²-(a²-b²-c²+l)²+m
z: c -> 2(2ab+m)(2ac+n)+n
If this ends up giving symmetry to one axis, it will be the unused z-axis. It would be straight forward to extend this to also treat the z-axis as "real" axis from time to time, which would push the iteration count one up.
If you prefer to do the two iterations seperately instead, you'll just have to add in some bit that knows, which type of iteration to take. The split iterations would be:
odd iterations:
x: a -> a²-b²-c²+l
y: b -> 2ab+m
z: c -> 2ac+n
even iterations:
x: a -> 2ab+l
y: b -> -a²+b²-c²+m
z: c -> 2bc+n
This should at least break the circle symmetry.
It's likely that something like this was tried too, already, but I can't quite recall.
It would also be easy to extend this to, say, a tetrahedral version, where you alternate over four different axes that correspond to a tetraheder.
What if you vary the rotational direction each time?
This is an example where, what would be the "real" axis alternates between the x-axis and the y-axis. Also, you're effectively doing two iterations with each iteration, because that's how I arrived at this closed form.
I differentiate the directions (x,y,z) from the coordinates (a,b,c) and the constant that is added, is denoted by (l,m,n):
x: a -> 2(a²-b²-c²+l)(2ab+m)+l
y: b -> (2ab+m)²-(2ac+n)²-(a²-b²-c²+l)²+m
z: c -> 2(2ab+m)(2ac+n)+n
If this ends up giving symmetry to one axis, it will be the unused z-axis. It would be straight forward to extend this to also treat the z-axis as "real" axis from time to time, which would push the iteration count one up.
If you prefer to do the two iterations seperately instead, you'll just have to add in some bit that knows, which type of iteration to take. The split iterations would be:
odd iterations:
x: a -> a²-b²-c²+l
y: b -> 2ab+m
z: c -> 2ac+n
even iterations:
x: a -> 2ab+l
y: b -> -a²+b²-c²+m
z: c -> 2bc+n
This should at least break the circle symmetry.
It's likely that something like this was tried too, already, but I can't quite recall.
It would also be easy to extend this to, say, a tetrahedral version, where you alternate over four different axes that correspond to a tetraheder.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on July 11, 2013, 08:41:48 AM
This is exactly falling into the same kind of reasoning I was working on in the quaternion thread. This is doubling the angle wrt the positive x direction & the positive y, which break the points out of being bound to a single plane.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on July 11, 2013, 11:19:25 AM
SEE: reformulation to avoid cancellation (http://www.fractalforums.com/new-theories-and-research/do-quaternions-need-revisiting/msg63511/#msg63511).
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on July 11, 2013, 11:38:15 AM
So your version tries to do this in a single iteration? Also nice.
I suppose, I could do this too, by just not adding the constant right away as well as choosing the square-root of a vector, rather than the vector itself... But something tells me, that would just be the same as having the whole thing be applied at an angle, rather than forcing two separate rotations. I might be wrong though.
Your version lools a bit different though.
I suppose, I could do this too, by just not adding the constant right away as well as choosing the square-root of a vector, rather than the vector itself... But something tells me, that would just be the same as having the whole thing be applied at an angle, rather than forcing two separate rotations. I might be wrong though.
Your version lools a bit different though.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on July 11, 2013, 02:36:02 PM
All I've done is reformed, say (x2-y2-z2) = (x+y+z)(x-y-z)+2yz to avoid the possibility of catastrophic cancellation and specify the direction of the line through the origin with respect the angle doubling is occurring. The linked equations are ignoring the translation part.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on July 11, 2013, 03:42:14 PM
yeah, ignoring the translation part is what I meant by doing it in one iteration
really nice work, now that there are examples of it all in the other thread ;P
really nice work, now that there are examples of it all in the other thread ;P
Title: Re: Geometric Algebra, Geometric Calculus
Post by: jehovajah on October 21, 2013, 01:13:41 AM
Excellent introduction to GA Kram.
I had a difficulty with the kind of mathematical Houdini approach of others, but you set it out for the most part sensibly and in a motivating way.
The Algebraic manipultions is where most non mathematicians get lost. Then the 1 to 1 isomorphisms( one thing in one notation is attached to only one thing in another notation, so we can say those things identify with each other, and the behaviours are the same for one notation as it is for the other notation) is the next place where people get lost.
Finally the lack of specificity in the term vector also loses people.
Honestly, each of these areas of loss and confusion I experienced reading your article, but because I knew what I was looking for I could see how you had tried to minimise these crisis points.
I learn something new each time I look at Grassmanns work, and I still think we over complicate it unnecessarily. The real problem is overcoming decades of bad teaching practices to be able to grasp the naturalness of these arithmetical concepts.
Nobody explains why the signature of a system is -1,0,1 because no one understands Grassmans dynamic model. I would highly recommend Norman Wildberger for all things Grasdmann even if he still gets some of it wrong, he is mostly right as far as the Grassmann method of Analysis is concerned. I have learned a lot through his clear exposition.
Grassmann defines Strecke as Die Begräntzte Linie, but also in the same context as the symbol for many discrete and continuous magnitudes in dynamic change. In this way he takes what is trigonometry of the parallelogram and the Triangle deconstructed into 3 points and 4 points with their associated stretch between them, into the subtleties of our interacting with space.
But what few grasp is that through Lagrange and his Father Justus, Grassmann was giving a version of Newtonian Analytics. And Newton was giving us his deep meditation on Pythagorean Mechanics and Astrological thought, as promulgated by the Platonic Academies.
Of those associated with the Euclidean branch of Plato's academy in Alexandria 2 stand out: Archimedes and Apollonius. But it is the work of Eudoxus that most clearly introduces the Pythagorean lineal algebra in the Logos Analogos format.
Making sense of how we interact with space before believing results of Algebraic manipultions is crucial to our sanity and sense of identity.
I had a difficulty with the kind of mathematical Houdini approach of others, but you set it out for the most part sensibly and in a motivating way.
The Algebraic manipultions is where most non mathematicians get lost. Then the 1 to 1 isomorphisms( one thing in one notation is attached to only one thing in another notation, so we can say those things identify with each other, and the behaviours are the same for one notation as it is for the other notation) is the next place where people get lost.
Finally the lack of specificity in the term vector also loses people.
Honestly, each of these areas of loss and confusion I experienced reading your article, but because I knew what I was looking for I could see how you had tried to minimise these crisis points.
I learn something new each time I look at Grassmanns work, and I still think we over complicate it unnecessarily. The real problem is overcoming decades of bad teaching practices to be able to grasp the naturalness of these arithmetical concepts.
Nobody explains why the signature of a system is -1,0,1 because no one understands Grassmans dynamic model. I would highly recommend Norman Wildberger for all things Grasdmann even if he still gets some of it wrong, he is mostly right as far as the Grassmann method of Analysis is concerned. I have learned a lot through his clear exposition.
Grassmann defines Strecke as Die Begräntzte Linie, but also in the same context as the symbol for many discrete and continuous magnitudes in dynamic change. In this way he takes what is trigonometry of the parallelogram and the Triangle deconstructed into 3 points and 4 points with their associated stretch between them, into the subtleties of our interacting with space.
But what few grasp is that through Lagrange and his Father Justus, Grassmann was giving a version of Newtonian Analytics. And Newton was giving us his deep meditation on Pythagorean Mechanics and Astrological thought, as promulgated by the Platonic Academies.
Of those associated with the Euclidean branch of Plato's academy in Alexandria 2 stand out: Archimedes and Apollonius. But it is the work of Eudoxus that most clearly introduces the Pythagorean lineal algebra in the Logos Analogos format.
Making sense of how we interact with space before believing results of Algebraic manipultions is crucial to our sanity and sense of identity.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: jehovajah on October 23, 2013, 11:58:33 AM
This is link to the Cayley contribution to Grassmans analytical method.
http://rsta.royalsocietypublishing.org/content/356/1740/1123.full.pdf
It takes a familiar Clifford algebra approach, and is too technical for my tastes, but if you have followed Norman and his video on area and volume in his WildLinAlg series you will recognise the point being put across.
How unfortunate that Mathematicians feel they have to bamboozle there audience with utter tripe while attempting to make a valuable contribution to methodology!
http://rsta.royalsocietypublishing.org/content/356/1740/1123.full.pdf
It takes a familiar Clifford algebra approach, and is too technical for my tastes, but if you have followed Norman and his video on area and volume in his WildLinAlg series you will recognise the point being put across.
How unfortunate that Mathematicians feel they have to bamboozle there audience with utter tripe while attempting to make a valuable contribution to methodology!
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on October 23, 2013, 02:14:27 PM
Years ago I skimmed the complete works of Cayley (available online)...it's quite interesting the range of topics and how various fields ended contributing to the development of linear algebra. While I'm babbling about old math papers I found it kind of interesting to see that "flame-wars" existed (at least) in the 19th century...they just did it in journals.
"What are you doing in there?"
"Writing an article...someone in Philosophical Magazine is wrong."
"What are you doing in there?"
"Writing an article...someone in Philosophical Magazine is wrong."
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on October 23, 2013, 04:57:03 PM
"What are you doing in there?"
"Writing an article...someone in Philosophical Magazine is wrong."
(http://imgs.xkcd.com/comics/duty_calls.png) (http://xkcd.com/386/)"Writing an article...someone in Philosophical Magazine is wrong."
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on October 23, 2013, 05:51:57 PM
I figured someone would get the reference.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: jehovajah on October 23, 2013, 06:48:36 PM
:rotfl:
Can you imagine, if they had the communication back then? Fast connections are not always a good thing! Lol!
Can you imagine, if they had the communication back then? Fast connections are not always a good thing! Lol!
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Nahee_Enterprises on October 24, 2013, 05:10:45 AM
Years ago I skimmed the complete works of Cayley (available online)... it's quite interesting the range of
topics and how various fields ended contributing to the development of linear algebra. While I'm babbling
about old math papers I found it kind of interesting to see that "flame-wars" existed (at least) in the 19th
century... they just did it in journals.
"What are you doing in there?"
"Writing an article...someone in Philosophical Magazine is wrong."
topics and how various fields ended contributing to the development of linear algebra. While I'm babbling
about old math papers I found it kind of interesting to see that "flame-wars" existed (at least) in the 19th
century... they just did it in journals.
"What are you doing in there?"
"Writing an article...someone in Philosophical Magazine is wrong."
http://imgs.xkcd.com/comics/duty_calls.png (http://xkcd.com/386/)
I figured someone would get the reference.
Yes, a very good reference. :stickingouttongue: :jabbering: Title: Re: Geometric Algebra, Geometric Calculus
Post by: Nahee_Enterprises on October 24, 2013, 05:12:56 AM
:rotfl:
Can you imagine, if they had the communication back then? Fast connections are not always a good thing! Lol!
Can you imagine, if they had the communication back then? Fast connections are not always a good thing! Lol!
Just wait and see what will happen when technology is able to directly link the brain to the computer for transferring thoughts immediately as they are thought. :D
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Dinkydau on October 25, 2013, 01:27:12 AM
I figured someone would get the reference.
HahaTitle: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on January 15, 2014, 02:37:33 PM
Alan Macdonald has an updated survey paper:
http://faculty.luther.edu/~macdonal/GA&GC.pdf
http://faculty.luther.edu/~macdonal/GA&GC.pdf
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on January 15, 2014, 06:12:15 PM
nice, thanks :)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on January 18, 2014, 02:54:10 AM
If I had to guess, this probably isn't quite legal, but I found this on google scholar:
Guide to Geometric Algebra in Practice - Leo Dorst, Daniel Fontijne and Stephen Mann (pdf) (http://f3.tiera.ru/2/M_Mathematics/MD_Geometry%20and%20topology/Dorst%20L.,%20Lasenby%20J.%20(eds.)%20Guide%20to%20geometric%20algebra%20in%20practice%20(Springer,%202011)(ISBN%200857298100)(O)(458s)_MD_.pdf)
Normally, you can only get a preview of it on Google books. It might not be up for long.
Guide to Geometric Algebra in Practice - Leo Dorst, Daniel Fontijne and Stephen Mann (pdf) (http://f3.tiera.ru/2/M_Mathematics/MD_Geometry%20and%20topology/Dorst%20L.,%20Lasenby%20J.%20(eds.)%20Guide%20to%20geometric%20algebra%20in%20practice%20(Springer,%202011)(ISBN%200857298100)(O)(458s)_MD_.pdf)
Normally, you can only get a preview of it on Google books. It might not be up for long.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: jehovajah on January 18, 2014, 03:34:08 PM
Good catch kram1032.
Sometimes they do make books available for free download, but springer are normally very tight!
Lol!
Sometimes they do make books available for free download, but springer are normally very tight!
Lol!
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on January 18, 2014, 11:21:08 PM
It's some russian server, judging from the url.
Well, have fun working through that :)
Well, have fun working through that :)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on February 16, 2014, 12:35:53 AM
another big summary of the topic:
Geometric Algebra: An Introduction with Applications in Euclidean and Conformal Geometry - Richard Alan Miller (http://scholarworks.sjsu.edu/cgi/viewcontent.cgi?article=7943&context=etd_theses)
Geometric Algebra: An Introduction with Applications in Euclidean and Conformal Geometry - Richard Alan Miller (http://scholarworks.sjsu.edu/cgi/viewcontent.cgi?article=7943&context=etd_theses)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on February 19, 2014, 05:10:57 AM
Hallo Kram1032,
thanks for the link to the paper of Alan Miller on the introduction on Geometric Algebra.
An exellent introduction into the subject. Easy to read and to understand.
thanks for the link to the paper of Alan Miller on the introduction on Geometric Algebra.
An exellent introduction into the subject. Easy to read and to understand.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on February 19, 2014, 11:41:23 AM
I have a feeling that, as more and more people understand and use geometric algebra, its grand summaries will become more and more comprehensible, which is great :)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Dinkydau on February 19, 2014, 05:46:53 PM
Hallo Kram1032,
thanks for the link to the paper of Alan Miller on the introduction on Geometric Algebra.
An exellent introduction into the subject. Easy to read and to understand.
Yes, this doesn't look so difficult as all those other scientific documents. Maybe even I could understand some of it.thanks for the link to the paper of Alan Miller on the introduction on Geometric Algebra.
An exellent introduction into the subject. Easy to read and to understand.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on February 22, 2014, 08:56:55 PM
Just in case you are not familiar with linear algebra: This is NOT a geometric algebra formulation of space. It's rather a 10-page crash-course-summary of the most important aspects of linear algebra. All that is done here can also be done in Geometric Algebra, though in many cases the Geometric Algebra formulation actually seems cleaner.
A Geometric Review of Linear Algebra (http://www.cns.nyu.edu/~eero/NOTES/geomLinAlg.pdf)
A Geometric Review of Linear Algebra (http://www.cns.nyu.edu/~eero/NOTES/geomLinAlg.pdf)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on February 23, 2014, 08:54:55 AM
Nice paper, I hope it is also readable for people who are not familiar with this subject.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on February 27, 2014, 02:41:23 AM
http://ryushare.me/dietmar-hildenbrand-foundations-of-geometric-algebra-computing-geometry-and-computing-pdf/#.Uw6Xgfl5PpE
another Springer book - a bit annoying to download though. Enjoy/be quick ;)
another Springer book - a bit annoying to download though. Enjoy/be quick ;)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: jehovajah on March 01, 2014, 11:20:46 AM
Thanks Kram1032.
This author or authors have read Grassmann I can tell from the first few opening remarks!
I think I will enjoy this approach because they understand the man and his background!
This author or authors have read Grassmann I can tell from the first few opening remarks!
I think I will enjoy this approach because they understand the man and his background!
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on March 20, 2014, 07:05:18 PM
Grassmann Algebra in Game Development (http://www.terathon.com/gdc14_lengyel.pdf) (slides from a talk at GDC'14)
It's interesting in that most other sources present geometric algebra in a very different way.
Usually, you first see the introduction of the geometric product of two vectors and, from there, a definition of the scalar and the wedge product.
In this one, they start with the wedge product and its dual form, the anti wedge product, and arrive at all the other things from there.
It also doesn't even mention the geometric product. I assume it's probably much closer to the formalisms of Grassmann than the later developments by Hestenes. But I'm not versed in the historical details. Clearly, this forum already has an expert on that front.
It's interesting in that most other sources present geometric algebra in a very different way.
Usually, you first see the introduction of the geometric product of two vectors and, from there, a definition of the scalar and the wedge product.
In this one, they start with the wedge product and its dual form, the anti wedge product, and arrive at all the other things from there.
It also doesn't even mention the geometric product. I assume it's probably much closer to the formalisms of Grassmann than the later developments by Hestenes. But I'm not versed in the historical details. Clearly, this forum already has an expert on that front.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on April 04, 2014, 12:05:18 AM
This very much isn't an introductory paper but it extends the ideas of conformal geometric algebra which has, as native objects, lines, planes, circles and spheres (for R³), to include quadratics or conics, e.g. arbitrary elipsoids, hyperboloids, paraboloids, cones and such things.
The idea is fairly simple, especially if you know a thing or two about conics/quadratics, but the presentation in this paper is, perhaps, a bit needlessly complex at times, at least imho.
Still, it's an interesting paper. With it you should be able to define "the" "elliptic"/"hyperbolic"/"parabolic" Mandelbrot set or a higher-dimensional analog.
(That higher-dimensional analog would most definitely just be a rotatory symmetric shape though. Although it might be a bit bendier than just a plain axial rotation, because the linear transformations in their conic Clifford Space do not represent linear transformations in Euklidean space. So things should *probably* distort a bit more...)
Anyway, here it is:
http://arxiv.org/pdf/1403.6665.pdf
The idea is fairly simple, especially if you know a thing or two about conics/quadratics, but the presentation in this paper is, perhaps, a bit needlessly complex at times, at least imho.
Still, it's an interesting paper. With it you should be able to define "the" "elliptic"/"hyperbolic"/"parabolic" Mandelbrot set or a higher-dimensional analog.
(That higher-dimensional analog would most definitely just be a rotatory symmetric shape though. Although it might be a bit bendier than just a plain axial rotation, because the linear transformations in their conic Clifford Space do not represent linear transformations in Euklidean space. So things should *probably* distort a bit more...)
Anyway, here it is:
http://arxiv.org/pdf/1403.6665.pdf
Title: Re: Geometric Algebra, Geometric Calculus
Post by: jehovajah on April 07, 2014, 11:39:35 AM
Once again Kram1032 thanks!
Yes it is a succinct exposition of the main differences between Grassman ,Clifford and Hamilton algebras. In a way it shows the kind of universal thinking of the time, in that Helmholtz attempted to combine the electromagnetic formulae of Maxwell and the Stokes / Navier equations into one by simply introducing an index factor. In this case the index is the choice of the square of terms !
I have, until now, always found Clifford's product distinctions confusing, but my research into Grassmanns concept has opened a can of worms! Well, I like worms and spiders, as you may know! :dink:
I have quite a long way to go in fully understanding Hermanns thinking, but I am grateful for small mercies. The recent translation was very revealing to me, because Hegel was unknown to me up to that point!
Thanks again and thank you for your research in this thread!
Yes it is a succinct exposition of the main differences between Grassman ,Clifford and Hamilton algebras. In a way it shows the kind of universal thinking of the time, in that Helmholtz attempted to combine the electromagnetic formulae of Maxwell and the Stokes / Navier equations into one by simply introducing an index factor. In this case the index is the choice of the square of terms !
I have, until now, always found Clifford's product distinctions confusing, but my research into Grassmanns concept has opened a can of worms! Well, I like worms and spiders, as you may know! :dink:
I have quite a long way to go in fully understanding Hermanns thinking, but I am grateful for small mercies. The recent translation was very revealing to me, because Hegel was unknown to me up to that point!
Thanks again and thank you for your research in this thread!
Title: Re: Geometric Algebra, Geometric Calculus
Post by: jehovajah on April 08, 2014, 07:27:51 AM
I have digested the slides and recognise them as pure Grassmann, excepting only the cross product. The cross product, despite it's failings, is still widely used and widely understood, so it merits treatment, but as he clearly points out, it is not up to the job! The example of a signed triangle and it's normal orientation is a case in point.
A normal orientation to a plane is still an important line segment, it is just not necessary for rotation!
Every aspect of these slides is exposited in detail by Norman Wildberger in his "Wild linear algebr"a series, except the anti vector, I think. That is unless he treats it under the kernel and basis videos which I have not fully explored.
This is a great Overview of Grassmanns thinking and the thinking of some of his followers in terms of the lineal algebra. The Schwenkung algebra of Grassmann would be interesting to see, but I think this is entirely developed within the Clifford algebras.
A normal orientation to a plane is still an important line segment, it is just not necessary for rotation!
Every aspect of these slides is exposited in detail by Norman Wildberger in his "Wild linear algebr"a series, except the anti vector, I think. That is unless he treats it under the kernel and basis videos which I have not fully explored.
This is a great Overview of Grassmanns thinking and the thinking of some of his followers in terms of the lineal algebra. The Schwenkung algebra of Grassmann would be interesting to see, but I think this is entirely developed within the Clifford algebras.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on April 13, 2014, 11:03:44 PM
As said, the previously posted paper allows for natively defining all conics within the propsed Geometric Algebra.
This makes reflection along any of the figures rather straight forward:
(http://puu.sh/87u5Y.png)
For circles, this is the commonly known sphere inversion. For ellipses, it's exactly the same, but it preserves the stretching. So I can only assume that an M-Set based on that would simply be a stretched M-set.
For Hyperbolas it becomes a little more interesting with the relationship between a line and its reflection being a bit less obvious.
Though the weirdest, to me, is the reflection of a line in a parabola: Note how for any of those reflections other than for the parabola, all the line-reflections go through the origin.
This is because the origin's inverse is the point at infinity. And of course any line meets at the point at infinity, so the reflection of any line must go through the origin.
Since that is not the case for a parabola, I wonder, for one thing, what significance the origin even has, as far as reflections are concerned, and for another: If we were to use the generalized M-set formula for even just the 2D case on this, would anything interesting happen? How would it look like?
Remember: The generic formula for squaring a number in GA is:
r² := r e r
where r is a vector and e is a unit vector which gives the direction the M-Set should point to. (The positive real axis in the original M-set)
So if you were to do this for a generic parabola and some unit vector, we should get "the" "parabolic" M-Set.
And I'm not even asking about the 3D M-Set resulting from that. It would most likely, once again, simply be a rotation of the 2D-Set. Although the above image suggests that, perhaps, it might have some interesting different behavior.
And even if not, at very least the 2D-M-Set resulting from this should act at least somewhat differently.
This makes reflection along any of the figures rather straight forward:
(http://puu.sh/87u5Y.png)
For circles, this is the commonly known sphere inversion. For ellipses, it's exactly the same, but it preserves the stretching. So I can only assume that an M-Set based on that would simply be a stretched M-set.
For Hyperbolas it becomes a little more interesting with the relationship between a line and its reflection being a bit less obvious.
Though the weirdest, to me, is the reflection of a line in a parabola: Note how for any of those reflections other than for the parabola, all the line-reflections go through the origin.
This is because the origin's inverse is the point at infinity. And of course any line meets at the point at infinity, so the reflection of any line must go through the origin.
Since that is not the case for a parabola, I wonder, for one thing, what significance the origin even has, as far as reflections are concerned, and for another: If we were to use the generalized M-set formula for even just the 2D case on this, would anything interesting happen? How would it look like?
Remember: The generic formula for squaring a number in GA is:
r² := r e r
where r is a vector and e is a unit vector which gives the direction the M-Set should point to. (The positive real axis in the original M-set)
So if you were to do this for a generic parabola and some unit vector, we should get "the" "parabolic" M-Set.
And I'm not even asking about the 3D M-Set resulting from that. It would most likely, once again, simply be a rotation of the 2D-Set. Although the above image suggests that, perhaps, it might have some interesting different behavior.
And even if not, at very least the 2D-M-Set resulting from this should act at least somewhat differently.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on April 13, 2014, 11:52:07 PM
Here is another paper covering this "Quadric Geometric Algebra" which might be a bit easier to understand. It certainly uses more traditional notation.
http://www.gaalop.de/wp-content/uploads/134-1061-Zamora.pdf
http://www.gaalop.de/wp-content/uploads/134-1061-Zamora.pdf
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on April 29, 2014, 02:20:58 PM
Here is a fine overview on Clifford Algebra/Geometric Algebra
http://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/ (http://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/)
GEOMETRIC ALGEBRA of one and many multivector variables
http://www.helsinki.fi/~jmpesone/index_files/GA.htm (http://www.helsinki.fi/~jmpesone/index_files/GA.htm)
Vectors, Spinors, and Complex Numbers in Classical and Quantum Physics - David Hestenes
http://geocalc.clas.asu.edu/pdf-preAdobe8/VeSp%26ComUpdated.pdf (http://geocalc.clas.asu.edu/pdf-preAdobe8/VeSp%26ComUpdated.pdf)
Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics
http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf (http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf) - David Hestenes
Spacetime Physics with Geometric Algebra - David Hestenes
http://geocalc.clas.asu.edu/pdf/SpacetimePhysics.pdf (http://geocalc.clas.asu.edu/pdf/SpacetimePhysics.pdf)
Gauge Theory Gravity with Geometric Calculus - David Hestenes
http://geocalc.clas.asu.edu/pdf/GTG.w.GC.FP.pdf (http://geocalc.clas.asu.edu/pdf/GTG.w.GC.FP.pdf)
http://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/ (http://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/)
GEOMETRIC ALGEBRA of one and many multivector variables
http://www.helsinki.fi/~jmpesone/index_files/GA.htm (http://www.helsinki.fi/~jmpesone/index_files/GA.htm)
Vectors, Spinors, and Complex Numbers in Classical and Quantum Physics - David Hestenes
http://geocalc.clas.asu.edu/pdf-preAdobe8/VeSp%26ComUpdated.pdf (http://geocalc.clas.asu.edu/pdf-preAdobe8/VeSp%26ComUpdated.pdf)
Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics
http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf (http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf) - David Hestenes
Spacetime Physics with Geometric Algebra - David Hestenes
http://geocalc.clas.asu.edu/pdf/SpacetimePhysics.pdf (http://geocalc.clas.asu.edu/pdf/SpacetimePhysics.pdf)
Gauge Theory Gravity with Geometric Calculus - David Hestenes
http://geocalc.clas.asu.edu/pdf/GTG.w.GC.FP.pdf (http://geocalc.clas.asu.edu/pdf/GTG.w.GC.FP.pdf)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on April 29, 2014, 11:57:48 PM
I feel like that blog post puts a bit too much magic behind the simple, straight forward idea of closure. And it really shouldn't be a mystery that i is a 90° rotation. That's already clear from how it works in complex numbers. Nor is it particularly strange that there is no difference between reflecting a single vector in the origin and 180° rotation of that vector around the origin. (If you consider more points, there actually is a difference. The same difference as being mirrored vs. being upside down. That seemed to be a little unclear from the exposition in that text)
All of those things, of course, are great to have in Clifford/Geometric Algebra. And some aspects are much easier to deal with and clearer therein. For instance, reflections that are NOT in the origin and rotations defined as double-reflections. (That's still not special in its own right but the corresponding notation in CA/GA is much more compact than the corresponding Vector Algebra analog)
Other than that, though, it's a nice read. Thanks for sharing :)
EDIT: Oh, you edited in a bunch of links. I am referring to the first.
As for the other links, I might not have linked them directly but I'm pretty sure I linked the corresponding pages that collected them.
All of those things, of course, are great to have in Clifford/Geometric Algebra. And some aspects are much easier to deal with and clearer therein. For instance, reflections that are NOT in the origin and rotations defined as double-reflections. (That's still not special in its own right but the corresponding notation in CA/GA is much more compact than the corresponding Vector Algebra analog)
Other than that, though, it's a nice read. Thanks for sharing :)
EDIT: Oh, you edited in a bunch of links. I am referring to the first.
As for the other links, I might not have linked them directly but I'm pretty sure I linked the corresponding pages that collected them.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on April 30, 2014, 09:32:53 AM
And it really shouldn't be a mystery that i is a 90° rotation.
I'd state this different. The 'i' of complex numbers (and generalizations) is the rule (and not a number) that makes the two numbers independent and how they interact over multiplication. The complex rule of i2 = -1 geometrically makes them orthogonal in a Euclidean sense and indirectly contains all of the rules of plane trigonometry. Change the rule for 'i' and you complete different notions of independence. Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on April 30, 2014, 10:13:12 PM
I'm not sure what you mean by that.
I was just referring to how slehar wrote up what he had to say about geometric algebra.
He makes some good points but much of it is neither specific to geometric algebra nor special.
That i = 90° rotation thing is pretty darn straight forward and should be taught in any semi-decent course on/ introduction to complex numbers.
What's really remarkable, and he did mention that too, is the seamless generalization of that idea to arbitrary dimensions. And a lot of objects to get a clearer, more straight forward description using these ideas. But that particular part of how the fact that i corresponds to a 90° rotation is rather mysterious seemed a bit far-fetched to me.
If you can't learn that from a course on complex numbers, a course on geometric algebra will probably not suddenly give you the epiphany to understand it.
I was just referring to how slehar wrote up what he had to say about geometric algebra.
He makes some good points but much of it is neither specific to geometric algebra nor special.
That i = 90° rotation thing is pretty darn straight forward and should be taught in any semi-decent course on/ introduction to complex numbers.
What's really remarkable, and he did mention that too, is the seamless generalization of that idea to arbitrary dimensions. And a lot of objects to get a clearer, more straight forward description using these ideas. But that particular part of how the fact that i corresponds to a 90° rotation is rather mysterious seemed a bit far-fetched to me.
If you can't learn that from a course on complex numbers, a course on geometric algebra will probably not suddenly give you the epiphany to understand it.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on May 01, 2014, 12:07:57 PM
I'll say the same thing in a different way. 'i' makes the two values linearly independent (same for the 'n' components of any system formed from a set of basis with no special rules for addition: clifford,quaternions,hypercomplex,etc)
In complex numbers the multiplication rule of 'i' causes multiplication to be the trig identity sum-of-angles (plus composition of scale) So multiplying by 'i' is a 90 degree rotation. Conjugation negates the angle and combined with multiply difference-of-angles (plus composition of scale). etc. Trig identities naturally fall out.
In complex numbers the multiplication rule of 'i' causes multiplication to be the trig identity sum-of-angles (plus composition of scale) So multiplying by 'i' is a 90 degree rotation. Conjugation negates the angle and combined with multiply difference-of-angles (plus composition of scale). etc. Trig identities naturally fall out.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on August 23, 2014, 12:28:12 PM
Another paper
Geometric Algebra for Electrical and Electronic Engineers (http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6876131)
As the title may suggest, this one focuses on electromagnetism. It's rather physics-heavy, but it looks nice.
Geometric Algebra for Electrical and Electronic Engineers (http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6876131)
As the title may suggest, this one focuses on electromagnetism. It's rather physics-heavy, but it looks nice.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: David Makin on August 24, 2014, 04:18:02 PM
While this thread's getting pretty heavy mathematically - does anyone know about proofs relating to mathematical fields in higher dimensions other than that it's proved that there's no R3 form that constitutes a field that isn't simply rewritable as an R2 form ?
e.g. is there a similar proof that says there are no mathematical fields in R4 or R5+ ?
All I know for sure is I can't find any true Rn fields documented for n>2.
e.g. is there a similar proof that says there are no mathematical fields in R4 or R5+ ?
All I know for sure is I can't find any true Rn fields documented for n>2.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on August 24, 2014, 04:32:34 PM
what do you mean by field? There are fields of any number of dimensions.
http://en.wikipedia.org/wiki/Field_(mathematics)
http://en.wikipedia.org/wiki/Field_(mathematics)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: David Makin on August 24, 2014, 04:58:01 PM
what do you mean by field? There are fields of any number of dimensions.
http://en.wikipedia.org/wiki/Field_(mathematics)
http://en.wikipedia.org/wiki/Field_(mathematics)
I mean infinite like reals and complex - not restrictive like p-adics etc.
Or perhaps I mean standard Euclidean ? Not sure, I'm not great on formal math ;)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on August 24, 2014, 07:12:37 PM
Hallo David,
the basic trick is, that you have algorithems to work with unit vectors. For example expand your mathematical universe with the following equations (relations between unit vectors):
With these definitions you can bring structure into complex numbers:
Define the complex value i in the following way:
now do squaring this equation (very detailed steps, inserting the above equations):
^2 = (e_1 e_2)(e_1 e_2)= -(e_2 e_1)(e_1 e_2) = -(e_2) (e_1 e_1) (e_2) = -(e_2)1(e_2) = -1 (e_2)(e_2) = -1 (e_2 e_2) = -1(1) = -1)
It is possible to redefine is this way also quaterions, octerions and other hyper complex numbers.
For the geometric interpretation (what are this unit vectors
) and as starter on this issue I propose the paper by Alan Miller:
http://scholarworks.sjsu.edu/cgi/viewcontent.cgi?article=7943&context=etd_theses (http://scholarworks.sjsu.edu/cgi/viewcontent.cgi?article=7943&context=etd_theses)
Look carefully on the geometric interpretation of the elements of this algebra. It opens a new way to do geometry
The paper of Alan Miller was for me an opener to other papers.
Hermann
the basic trick is, that you have algorithems to work with unit vectors. For example expand your mathematical universe with the following equations (relations between unit vectors):
With these definitions you can bring structure into complex numbers:
Define the complex value i in the following way:
now do squaring this equation (very detailed steps, inserting the above equations):
It is possible to redefine is this way also quaterions, octerions and other hyper complex numbers.
For the geometric interpretation (what are this unit vectors
http://scholarworks.sjsu.edu/cgi/viewcontent.cgi?article=7943&context=etd_theses (http://scholarworks.sjsu.edu/cgi/viewcontent.cgi?article=7943&context=etd_theses)
Look carefully on the geometric interpretation of the elements of this algebra. It opens a new way to do geometry
The paper of Alan Miller was for me an opener to other papers.
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on August 24, 2014, 07:13:43 PM
Do you mean division ring?
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on August 24, 2014, 07:23:06 PM
Do you mean division ring?
I don't know what a division ring is. I talk about geometric algebra. May be it is a division ring.
In geometric algebra is is possible doing devision by vectors!
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on August 24, 2014, 09:55:51 PM
That was to David. It sounded to me like he's asking about Frobenius theorem.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on August 25, 2014, 09:28:03 PM
@David
p-adics are infinite just like the reals are. In fact, there is only one kind of real but there are countably infinitely many p-adics (one per prime number)
And standard-Euclidean...
I think you'll have to clear up your own vocabulary there, because that makes little sense in the context :)
@hermann
a division ring is a ring that also has a multiplicative inverse.
A ring is some kind of number structure which has a form of addition and a form of multiplication which includes the classical rules:
addition is commutative and associative, multiplication is associative, and together, they are distributive; addition has the neutral element 0 and multiplication has the neutral element 1. Addition also has its inverse, while multiplication doesn't necessarily have to have that.
Examples:
the "classical" algebra over R is a commutative division ring.
R³-vector algebra with the cross product is an anti-commutative ring.
Geometric Algebras are division-rings.
etc.
p-adics are infinite just like the reals are. In fact, there is only one kind of real but there are countably infinitely many p-adics (one per prime number)
And standard-Euclidean...
I think you'll have to clear up your own vocabulary there, because that makes little sense in the context :)
@hermann
a division ring is a ring that also has a multiplicative inverse.
A ring is some kind of number structure which has a form of addition and a form of multiplication which includes the classical rules:
addition is commutative and associative, multiplication is associative, and together, they are distributive; addition has the neutral element 0 and multiplication has the neutral element 1. Addition also has its inverse, while multiplication doesn't necessarily have to have that.
Examples:
the "classical" algebra over R is a commutative division ring.
R³-vector algebra with the cross product is an anti-commutative ring.
Geometric Algebras are division-rings.
etc.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on August 26, 2014, 07:46:52 AM
Thanks kram1032,
I expected that Geometric Algebra is a division ring.
p-adic numbers.
http://en.wikipedia.org/wiki/P-adic_number (http://en.wikipedia.org/wiki/P-adic_number)
On a computer I prefer to have a number writen in the form
(Rationa Number) instead of 0.333333333333333333333333333333333333333333333333333333333333333333333333...
Three dots ... mean go on to infinity. The first presentation has also the advatage to require less storage space on a computer.
A decimal presentation on a computer means that one has to cut digits and loses precision.
Prime Factorization
Integers can be then be split into prime factors:
1482 = 2 * 3 * 13 * 19
1488 = 2 * 2 * 2 * 2 * 3 * 31
http://www.wackerart.de/mathematik/primfaktoren_zerlegung.html (http://www.wackerart.de/mathematik/primfaktoren_zerlegung.html)
Devision can then be made much easier when one has stored integers in the form of prime products and as rational numbers only.
The following form can be implemented very effectively on a computer.
Donald E. Knuth has worked this out in detail in one of his books on "The Art of Computer Programming".
I tried to implement a package for rational number operations but then discovered that intergers implemented on a computer are limited.
So I need to implement a package of unbounded integers to make it work well.
(I know that such implementaions already exist but I prefer to program it by my self. It is for me more fun to program instead of understanding code of other programmers)
Hermann
P.S Thanks for the inspiration.
I expected that Geometric Algebra is a division ring.
p-adic numbers.
http://en.wikipedia.org/wiki/P-adic_number (http://en.wikipedia.org/wiki/P-adic_number)
On a computer I prefer to have a number writen in the form
Three dots ... mean go on to infinity. The first presentation has also the advatage to require less storage space on a computer.
A decimal presentation on a computer means that one has to cut digits and loses precision.
Prime Factorization
Integers can be then be split into prime factors:
1482 = 2 * 3 * 13 * 19
1488 = 2 * 2 * 2 * 2 * 3 * 31
http://www.wackerart.de/mathematik/primfaktoren_zerlegung.html (http://www.wackerart.de/mathematik/primfaktoren_zerlegung.html)
Devision can then be made much easier when one has stored integers in the form of prime products and as rational numbers only.
The following form can be implemented very effectively on a computer.
Donald E. Knuth has worked this out in detail in one of his books on "The Art of Computer Programming".
I tried to implement a package for rational number operations but then discovered that intergers implemented on a computer are limited.
So I need to implement a package of unbounded integers to make it work well.
(I know that such implementaions already exist but I prefer to program it by my self. It is for me more fun to program instead of understanding code of other programmers)
Hermann
P.S Thanks for the inspiration.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on August 26, 2014, 12:51:45 PM
hermann you might want to look up "computable numbers" or "limit-computable numbers" ;)
Limit-computable numbers are the biggest countable set of numbers. It's not quite the reals, but it's effectively all the reals (with few exceptions) you'd ever care about.
That being said, this is getting rather off topic now. None of this has to do specifically with Geometric Algebra or Geometric Calculus.
Limit-computable numbers are the biggest countable set of numbers. It's not quite the reals, but it's effectively all the reals (with few exceptions) you'd ever care about.
That being said, this is getting rather off topic now. None of this has to do specifically with Geometric Algebra or Geometric Calculus.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on August 26, 2014, 03:24:36 PM
That being said, this is getting rather off topic now. None of this has to do specifically with Geometric Algebra or Geometric Calculus.
Hallo Kram1032,
you know I am playing a bit arround with trying to implement a generic software for geometric algebra.
In preperation I had to implement the n over k algorithem. As preperation for n over k I had to implement the factorial function.
A short test programm soon showed me that one comes to the frontiers for the implementation of integer on a computer very fast:
Factorial:
0!: 1
1!: 1
2!: 2
3!: 6
4!: 24
5!: 120
6!: 720
7!: 5040
8!: 40320
9!: 362880
10!: 3628800
11!: 39916800
12!: 479001600
until know it's ok but then it goes wrong!
13!: 1932053504
14!: 1278945280
n over k (binominals)
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
1 11 55 165 330 462 462 330 165 55 11 1
1 12 66 220 495 792 924 792 495 220 66 12 1
The next step goes wrong!
1 13 78 286 715 1287 1716 1716 1287 715 286 -29 4 1
1 14 91 364 1001 2002 3003 3432 3003 2002 -182 41 1 0 1
That means using geometric algebra in higher dimensions increases the efford rapidly!
I also had expected, that my program would throw an exception caused by this numerical overflow, but it does not!
Which is another pitfall for the implementation.
The idear of implementig unbounded integer I had in mind for a long time.
(I know that this has already been done!)
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 01, 2014, 01:17:20 PM
I've never used Perl but here is a GA implementation in Perl 6 if anybody is interested in that
https://github.com/grondilu/clifford
https://github.com/grondilu/clifford
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on September 01, 2014, 01:46:47 PM
hermann: focus on the tree. Don't worry about any over 5. If five some ends up being too small then deal with that went it comes.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 01, 2014, 03:16:43 PM
since those numbers don't actually grow all that fast, you could also define n over k recursively via integer addition, using a "dictionary" (a dynamically extended lookup table) to keep that implementation reasonably efficient without using big num operations.
Just define a bunch of relations you'll need:Technically you only need the base case (0,0), the out of bounds case k<0 or k>n, and the recursive addition to really catch them all, but the other couple rules up there are so simple, I don't see why you wouldn't add them in anyway.
A naive implementation of this will suffer from the same issues, a naive implementation of recursive Fibonacci numbers or a naive recursive factorial would suffer from, so if you go this path, I highly recommend dynamically generating a lookup table as you go, to keep down calculation times.
If you are concerned about the memory consumption of such a table (which you really shouldn't be - it shouldn't be a big deal to store numbers of those sizes), you can employ one extra rule to essentially half that memory consumption by using the rule
All this together should give you a pretty acceptable, reasonably fast implementation of Binomial(n,k) - The largest of Binomial coefficients still grows rather quickly, though. With your standard unsigned (32-bit) int precision, you'd get up to Binomial(18,9) = 48620 with this. Binomial(19,k) will start to see problems once again. However, you'll find yourself pretty rarely in a situation where you actually need to use anything of dimension 19.
The largest Binomial Coefficient grows with roughly
, while the general asymptotic growth of any binomial coefficient is about ^{k-n-\frac{1}{2}})
.
Another thing you could try is to figure out precisely what multiplications you do not actually care about, to also avoid overflow.
12!/(6! (12-6)!) = (12*11*10*9*8*7*6*5*4*3*2)/((6*5*4*3*2)*(6*5*4*3*2)) =
(12*11*10*9*8*7)/(6*5*4*3*2) =
(2*11*2*3*1*7) = 11*7*3*22 = 924
This would require some kind of implementation of rationals where you can reduce them to their simplest form symbolically, before even executing any numeric multiplication. That could be interesting and worthwhile in its own right but it's probably also beyond what you are trying to do here. In the end, it would probably be slower and it'd suffer from the same limitations as the recursive definition above.
All that being said, I'm with Roquen that 12! is more than sufficient for your needs. Just work on other parts. If needed, you can convert to bignum later down the line just fine.
Appendix: Full Form of the Asymptotic Behavior of the Largest Binomial Coefficient, Corrected for Even and Odd Values for n:
^{-\lceil \frac{n}{2}\rceil -\frac{1}{2}} \left(\lfloor \frac{n}{2}\rfloor +1\right)^{-\lfloor \frac{n}{2}\rfloor -\frac{1}{2}} \: e^{\frac{1}{360} \left(-\frac{30}{\lceil \frac{n}{2}\rceil +1}+\frac{1}{\left(\lceil\frac{n}{2}\rceil +1\right)^3}+360 \lfloor \frac{n}{2}\rfloor -\frac{30}{\lfloor \frac{n}{2}\rfloor +1}+\frac{1}{\left(\lfloor\frac{n}{2}\rfloor +1\right)^3}-360 n+720\right)+\lceil \frac{n}{2}\rceil } )
Just define a bunch of relations you'll need:
Code:
binomial(0,0) = 1 // base case
// (1,k) for all k - you could even ignore the base case since it also is defined by this
binomial(n,0) = 1
binomial(n,n) = 1
//(2,k) and (3,k) for all k
binomial(n,1) = n
binomial(n,n-1) = n
//out of bounds binomial coefficients
if k<0 || k>n => binomial(n,k) = 0
//all other possible cases, without any multiplication at all
binomial(n,k) = binomial(n-1,k-1) + binomial(n-1,k)
A naive implementation of this will suffer from the same issues, a naive implementation of recursive Fibonacci numbers or a naive recursive factorial would suffer from, so if you go this path, I highly recommend dynamically generating a lookup table as you go, to keep down calculation times.
If you are concerned about the memory consumption of such a table (which you really shouldn't be - it shouldn't be a big deal to store numbers of those sizes), you can employ one extra rule to essentially half that memory consumption by using the rule
Code:
If k>n/2 => Binomial(n,k) = Binomial(n,n-k)
The largest Binomial Coefficient grows with roughly
Another thing you could try is to figure out precisely what multiplications you do not actually care about, to also avoid overflow.
12!/(6! (12-6)!) = (12*11*10*9*8*7*6*5*4*3*2)/((6*5*4*3*2)*(6*5*4*3*2)) =
(12*11*10*9*8*7)/(6*5*4*3*2) =
(2*11*2*3*1*7) = 11*7*3*22 = 924
This would require some kind of implementation of rationals where you can reduce them to their simplest form symbolically, before even executing any numeric multiplication. That could be interesting and worthwhile in its own right but it's probably also beyond what you are trying to do here. In the end, it would probably be slower and it'd suffer from the same limitations as the recursive definition above.
All that being said, I'm with Roquen that 12! is more than sufficient for your needs. Just work on other parts. If needed, you can convert to bignum later down the line just fine.
Appendix: Full Form of the Asymptotic Behavior of the Largest Binomial Coefficient, Corrected for Even and Odd Values for n:
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on September 02, 2014, 03:25:28 PM
Hallo Krams, thanks for the lengthy formula!
I didn't want to go to much in the details.
I was only astonisched how fast one gets to the limits of the computer when callculating n_over_k!
With my Program I can know generate the base vectors here is an example for a space with dimension 4.
The LaTex table is generated by the Program.
I can post higher dimensions if one likes!
I didn't want to go to much in the details.
I was only astonisched how fast one gets to the limits of the computer when callculating n_over_k!
With my Program I can know generate the base vectors here is an example for a space with dimension 4.
The LaTex table is generated by the Program.
I can post higher dimensions if one likes!
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on September 02, 2014, 03:42:06 PM
The number of base elements raises rapidly with the dimension a bit higher!
Here is Dimension 6!
Use the bottom scrollbar to see the full table!
Here is Dimension 6!
Use the bottom scrollbar to see the full table!
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 02, 2014, 04:08:19 PM
Well yeah: an algebra of dimension 6 has 26=64 different elements.
I'd suggest naming the sole full-dimensional element I and defining the later half by multiplying I with the lower half.
This gives a fairly intuitive relationship between various elements and their respective orthogonality.
For instance, you will find that, in the case of R3:
I e1 = e1e2e3 e1 = e2e3
I e2 = e1e2e3 e2 = -e1e3 = e3e1
I e3 = e1e2e3 e3 = e1e2
So the "1st bivector" would be e2e3 rather than e1e2, and the bivector "e1e3" would actually appear as negative by default, replacing it by "e3e1" as the positive one. - For a right-handed frame, rather than a left-handed one.
If you use this ordering in, say, multiplication tables, you will find that it is more natural than the ordering you currently use in your tables. There will be more visible, obvious structure.
This will also be useful in defining relationships over dual spaces, like the geometrically very useful meet and join operations.
Edit: Actually, scratch that. This definitely works perfectly fine with 3 base elements. But beyond that, the ordering isn't so obvious. I'm not sure if there is some kind of "ideal" "canonical" ordering in that case.
I'd suggest naming the sole full-dimensional element I and defining the later half by multiplying I with the lower half.
This gives a fairly intuitive relationship between various elements and their respective orthogonality.
For instance, you will find that, in the case of R3:
I e1 = e1e2e3 e1 = e2e3
I e2 = e1e2e3 e2 = -e1e3 = e3e1
I e3 = e1e2e3 e3 = e1e2
So the "1st bivector" would be e2e3 rather than e1e2, and the bivector "e1e3" would actually appear as negative by default, replacing it by "e3e1" as the positive one. - For a right-handed frame, rather than a left-handed one.
If you use this ordering in, say, multiplication tables, you will find that it is more natural than the ordering you currently use in your tables. There will be more visible, obvious structure.
This will also be useful in defining relationships over dual spaces, like the geometrically very useful meet and join operations.
Edit: Actually, scratch that. This definitely works perfectly fine with 3 base elements. But beyond that, the ordering isn't so obvious. I'm not sure if there is some kind of "ideal" "canonical" ordering in that case.
Title: Canonic Base
Post by: hermann on September 03, 2014, 06:11:41 AM
Hallo Kram,
thanks for the posting of the link to the paper of Daniel Fontijne on Efficient Implementation of Geometric Algebra.
This paper was a great inspiration for my work!
Fontijne speeks from canonic ordering. I have implemented it a bit different then proposed by Fontijne (it was easier to program).
But it should produce the same result for the canonic ordering and labeling of the indizies. The trick is to implement the Base by binary numbers!
From the binary base it is easy to transform to the name of the element. The grade is the number of 1's in the binary representation.
For the principal have a look at the 6-Dimensional output.
I had some trouble with all the indizies. But now I have the basic data structures and subroutines for the initialisation of all the indizies.
I also have some subroutines for producing LaTex- and HTML-tables for the visualisation.
Implementing the product of geometric algebra shouldn't be much effort now.
Fontijne has given an example, but I have to modify it to match my data stuctures.
I hope I have some time for this issues the next days.
Hermann
P.S The following output shows the unit vectors for 6 Dimensions.
Ps. I had problems in placing the LaTex Table for Dimension 8. may be to many elements!
thanks for the posting of the link to the paper of Daniel Fontijne on Efficient Implementation of Geometric Algebra.
This paper was a great inspiration for my work!
Fontijne speeks from canonic ordering. I have implemented it a bit different then proposed by Fontijne (it was easier to program).
But it should produce the same result for the canonic ordering and labeling of the indizies. The trick is to implement the Base by binary numbers!
From the binary base it is easy to transform to the name of the element. The grade is the number of 1's in the binary representation.
For the principal have a look at the 6-Dimensional output.
I had some trouble with all the indizies. But now I have the basic data structures and subroutines for the initialisation of all the indizies.
I also have some subroutines for producing LaTex- and HTML-tables for the visualisation.
Implementing the product of geometric algebra shouldn't be much effort now.
Fontijne has given an example, but I have to modify it to match my data stuctures.
I hope I have some time for this issues the next days.
Hermann
P.S The following output shows the unit vectors for 6 Dimensions.
Code:
0: Value: 1.00000E+00 Grade: 0 Base: 0 0 0 0 0 0 Name: 1
1: Value: 1.00000E+00 Grade: 1 Base: 1 0 0 0 0 0 Name: e1
2: Value: 1.00000E+00 Grade: 1 Base: 0 1 0 0 0 0 Name: e2
3: Value: 1.00000E+00 Grade: 2 Base: 1 1 0 0 0 0 Name: e1^e2
4: Value: 1.00000E+00 Grade: 1 Base: 0 0 1 0 0 0 Name: e3
5: Value: 1.00000E+00 Grade: 2 Base: 1 0 1 0 0 0 Name: e1^e3
6: Value: 1.00000E+00 Grade: 2 Base: 0 1 1 0 0 0 Name: e2^e3
7: Value: 1.00000E+00 Grade: 3 Base: 1 1 1 0 0 0 Name: e1^e2^e3
8: Value: 1.00000E+00 Grade: 1 Base: 0 0 0 1 0 0 Name: e4
9: Value: 1.00000E+00 Grade: 2 Base: 1 0 0 1 0 0 Name: e1^e4
10: Value: 1.00000E+00 Grade: 2 Base: 0 1 0 1 0 0 Name: e2^e4
11: Value: 1.00000E+00 Grade: 3 Base: 1 1 0 1 0 0 Name: e1^e2^e4
12: Value: 1.00000E+00 Grade: 2 Base: 0 0 1 1 0 0 Name: e3^e4
13: Value: 1.00000E+00 Grade: 3 Base: 1 0 1 1 0 0 Name: e1^e3^e4
14: Value: 1.00000E+00 Grade: 3 Base: 0 1 1 1 0 0 Name: e2^e3^e4
15: Value: 1.00000E+00 Grade: 4 Base: 1 1 1 1 0 0 Name: e1^e2^e3^e4
16: Value: 1.00000E+00 Grade: 1 Base: 0 0 0 0 1 0 Name: e5
17: Value: 1.00000E+00 Grade: 2 Base: 1 0 0 0 1 0 Name: e1^e5
18: Value: 1.00000E+00 Grade: 2 Base: 0 1 0 0 1 0 Name: e2^e5
19: Value: 1.00000E+00 Grade: 3 Base: 1 1 0 0 1 0 Name: e1^e2^e5
20: Value: 1.00000E+00 Grade: 2 Base: 0 0 1 0 1 0 Name: e3^e5
21: Value: 1.00000E+00 Grade: 3 Base: 1 0 1 0 1 0 Name: e1^e3^e5
22: Value: 1.00000E+00 Grade: 3 Base: 0 1 1 0 1 0 Name: e2^e3^e5
23: Value: 1.00000E+00 Grade: 4 Base: 1 1 1 0 1 0 Name: e1^e2^e3^e5
24: Value: 1.00000E+00 Grade: 2 Base: 0 0 0 1 1 0 Name: e4^e5
25: Value: 1.00000E+00 Grade: 3 Base: 1 0 0 1 1 0 Name: e1^e4^e5
26: Value: 1.00000E+00 Grade: 3 Base: 0 1 0 1 1 0 Name: e2^e4^e5
27: Value: 1.00000E+00 Grade: 4 Base: 1 1 0 1 1 0 Name: e1^e2^e4^e5
28: Value: 1.00000E+00 Grade: 3 Base: 0 0 1 1 1 0 Name: e3^e4^e5
29: Value: 1.00000E+00 Grade: 4 Base: 1 0 1 1 1 0 Name: e1^e3^e4^e5
30: Value: 1.00000E+00 Grade: 4 Base: 0 1 1 1 1 0 Name: e2^e3^e4^e5
31: Value: 1.00000E+00 Grade: 5 Base: 1 1 1 1 1 0 Name: e1^e2^e3^e4^e5
32: Value: 1.00000E+00 Grade: 1 Base: 0 0 0 0 0 1 Name: e6
33: Value: 1.00000E+00 Grade: 2 Base: 1 0 0 0 0 1 Name: e1^e6
34: Value: 1.00000E+00 Grade: 2 Base: 0 1 0 0 0 1 Name: e2^e6
35: Value: 1.00000E+00 Grade: 3 Base: 1 1 0 0 0 1 Name: e1^e2^e6
36: Value: 1.00000E+00 Grade: 2 Base: 0 0 1 0 0 1 Name: e3^e6
37: Value: 1.00000E+00 Grade: 3 Base: 1 0 1 0 0 1 Name: e1^e3^e6
38: Value: 1.00000E+00 Grade: 3 Base: 0 1 1 0 0 1 Name: e2^e3^e6
39: Value: 1.00000E+00 Grade: 4 Base: 1 1 1 0 0 1 Name: e1^e2^e3^e6
40: Value: 1.00000E+00 Grade: 2 Base: 0 0 0 1 0 1 Name: e4^e6
41: Value: 1.00000E+00 Grade: 3 Base: 1 0 0 1 0 1 Name: e1^e4^e6
42: Value: 1.00000E+00 Grade: 3 Base: 0 1 0 1 0 1 Name: e2^e4^e6
43: Value: 1.00000E+00 Grade: 4 Base: 1 1 0 1 0 1 Name: e1^e2^e4^e6
44: Value: 1.00000E+00 Grade: 3 Base: 0 0 1 1 0 1 Name: e3^e4^e6
45: Value: 1.00000E+00 Grade: 4 Base: 1 0 1 1 0 1 Name: e1^e3^e4^e6
46: Value: 1.00000E+00 Grade: 4 Base: 0 1 1 1 0 1 Name: e2^e3^e4^e6
47: Value: 1.00000E+00 Grade: 5 Base: 1 1 1 1 0 1 Name: e1^e2^e3^e4^e6
48: Value: 1.00000E+00 Grade: 2 Base: 0 0 0 0 1 1 Name: e5^e6
49: Value: 1.00000E+00 Grade: 3 Base: 1 0 0 0 1 1 Name: e1^e5^e6
50: Value: 1.00000E+00 Grade: 3 Base: 0 1 0 0 1 1 Name: e2^e5^e6
51: Value: 1.00000E+00 Grade: 4 Base: 1 1 0 0 1 1 Name: e1^e2^e5^e6
52: Value: 1.00000E+00 Grade: 3 Base: 0 0 1 0 1 1 Name: e3^e5^e6
53: Value: 1.00000E+00 Grade: 4 Base: 1 0 1 0 1 1 Name: e1^e3^e5^e6
54: Value: 1.00000E+00 Grade: 4 Base: 0 1 1 0 1 1 Name: e2^e3^e5^e6
55: Value: 1.00000E+00 Grade: 5 Base: 1 1 1 0 1 1 Name: e1^e2^e3^e5^e6
56: Value: 1.00000E+00 Grade: 3 Base: 0 0 0 1 1 1 Name: e4^e5^e6
57: Value: 1.00000E+00 Grade: 4 Base: 1 0 0 1 1 1 Name: e1^e4^e5^e6
58: Value: 1.00000E+00 Grade: 4 Base: 0 1 0 1 1 1 Name: e2^e4^e5^e6
59: Value: 1.00000E+00 Grade: 5 Base: 1 1 0 1 1 1 Name: e1^e2^e4^e5^e6
60: Value: 1.00000E+00 Grade: 4 Base: 0 0 1 1 1 1 Name: e3^e4^e5^e6
61: Value: 1.00000E+00 Grade: 5 Base: 1 0 1 1 1 1 Name: e1^e3^e4^e5^e6
62: Value: 1.00000E+00 Grade: 5 Base: 0 1 1 1 1 1 Name: e2^e3^e4^e5^e6
63: Value: 1.00000E+00 Grade: 6 Base: 1 1 1 1 1 1 Name: e1^e2^e3^e4^e5^e6
Ps. I had problems in placing the LaTex Table for Dimension 8. may be to many elements!
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on September 03, 2014, 06:41:08 AM
I think the principle is easy to understand in three Dimensions:
Understood ? Here we have 8 Dimensions:
Quote
0: Value: 1.00000E+00 Grade: 0 Base: 0 0 0 Name: 1
1: Value: 1.00000E+00 Grade: 1 Base: 1 0 0 Name: e1
2: Value: 1.00000E+00 Grade: 1 Base: 0 1 0 Name: e2
3: Value: 1.00000E+00 Grade: 2 Base: 1 1 0 Name: e1^e2
4: Value: 1.00000E+00 Grade: 1 Base: 0 0 1 Name: e3
5: Value: 1.00000E+00 Grade: 2 Base: 1 0 1 Name: e1^e3
6: Value: 1.00000E+00 Grade: 2 Base: 0 1 1 Name: e2^e3
7: Value: 1.00000E+00 Grade: 3 Base: 1 1 1 Name: e1^e2^e3
1: Value: 1.00000E+00 Grade: 1 Base: 1 0 0 Name: e1
2: Value: 1.00000E+00 Grade: 1 Base: 0 1 0 Name: e2
3: Value: 1.00000E+00 Grade: 2 Base: 1 1 0 Name: e1^e2
4: Value: 1.00000E+00 Grade: 1 Base: 0 0 1 Name: e3
5: Value: 1.00000E+00 Grade: 2 Base: 1 0 1 Name: e1^e3
6: Value: 1.00000E+00 Grade: 2 Base: 0 1 1 Name: e2^e3
7: Value: 1.00000E+00 Grade: 3 Base: 1 1 1 Name: e1^e2^e3
Understood ? Here we have 8 Dimensions:
Code:
0: Value: 1.00000E+00 Grade: 0 Base: 0 0 0 0 0 0 0 0 Name: 1
1: Value: 1.00000E+00 Grade: 1 Base: 1 0 0 0 0 0 0 0 Name: e1
2: Value: 1.00000E+00 Grade: 1 Base: 0 1 0 0 0 0 0 0 Name: e2
3: Value: 1.00000E+00 Grade: 2 Base: 1 1 0 0 0 0 0 0 Name: e1^e2
4: Value: 1.00000E+00 Grade: 1 Base: 0 0 1 0 0 0 0 0 Name: e3
5: Value: 1.00000E+00 Grade: 2 Base: 1 0 1 0 0 0 0 0 Name: e1^e3
6: Value: 1.00000E+00 Grade: 2 Base: 0 1 1 0 0 0 0 0 Name: e2^e3
7: Value: 1.00000E+00 Grade: 3 Base: 1 1 1 0 0 0 0 0 Name: e1^e2^e3
8: Value: 1.00000E+00 Grade: 1 Base: 0 0 0 1 0 0 0 0 Name: e4
9: Value: 1.00000E+00 Grade: 2 Base: 1 0 0 1 0 0 0 0 Name: e1^e4
10: Value: 1.00000E+00 Grade: 2 Base: 0 1 0 1 0 0 0 0 Name: e2^e4
11: Value: 1.00000E+00 Grade: 3 Base: 1 1 0 1 0 0 0 0 Name: e1^e2^e4
12: Value: 1.00000E+00 Grade: 2 Base: 0 0 1 1 0 0 0 0 Name: e3^e4
13: Value: 1.00000E+00 Grade: 3 Base: 1 0 1 1 0 0 0 0 Name: e1^e3^e4
14: Value: 1.00000E+00 Grade: 3 Base: 0 1 1 1 0 0 0 0 Name: e2^e3^e4
15: Value: 1.00000E+00 Grade: 4 Base: 1 1 1 1 0 0 0 0 Name: e1^e2^e3^e4
16: Value: 1.00000E+00 Grade: 1 Base: 0 0 0 0 1 0 0 0 Name: e5
17: Value: 1.00000E+00 Grade: 2 Base: 1 0 0 0 1 0 0 0 Name: e1^e5
18: Value: 1.00000E+00 Grade: 2 Base: 0 1 0 0 1 0 0 0 Name: e2^e5
19: Value: 1.00000E+00 Grade: 3 Base: 1 1 0 0 1 0 0 0 Name: e1^e2^e5
20: Value: 1.00000E+00 Grade: 2 Base: 0 0 1 0 1 0 0 0 Name: e3^e5
21: Value: 1.00000E+00 Grade: 3 Base: 1 0 1 0 1 0 0 0 Name: e1^e3^e5
22: Value: 1.00000E+00 Grade: 3 Base: 0 1 1 0 1 0 0 0 Name: e2^e3^e5
23: Value: 1.00000E+00 Grade: 4 Base: 1 1 1 0 1 0 0 0 Name: e1^e2^e3^e5
24: Value: 1.00000E+00 Grade: 2 Base: 0 0 0 1 1 0 0 0 Name: e4^e5
25: Value: 1.00000E+00 Grade: 3 Base: 1 0 0 1 1 0 0 0 Name: e1^e4^e5
26: Value: 1.00000E+00 Grade: 3 Base: 0 1 0 1 1 0 0 0 Name: e2^e4^e5
27: Value: 1.00000E+00 Grade: 4 Base: 1 1 0 1 1 0 0 0 Name: e1^e2^e4^e5
28: Value: 1.00000E+00 Grade: 3 Base: 0 0 1 1 1 0 0 0 Name: e3^e4^e5
29: Value: 1.00000E+00 Grade: 4 Base: 1 0 1 1 1 0 0 0 Name: e1^e3^e4^e5
30: Value: 1.00000E+00 Grade: 4 Base: 0 1 1 1 1 0 0 0 Name: e2^e3^e4^e5
31: Value: 1.00000E+00 Grade: 5 Base: 1 1 1 1 1 0 0 0 Name: e1^e2^e3^e4^e5
32: Value: 1.00000E+00 Grade: 1 Base: 0 0 0 0 0 1 0 0 Name: e6
33: Value: 1.00000E+00 Grade: 2 Base: 1 0 0 0 0 1 0 0 Name: e1^e6
34: Value: 1.00000E+00 Grade: 2 Base: 0 1 0 0 0 1 0 0 Name: e2^e6
35: Value: 1.00000E+00 Grade: 3 Base: 1 1 0 0 0 1 0 0 Name: e1^e2^e6
36: Value: 1.00000E+00 Grade: 2 Base: 0 0 1 0 0 1 0 0 Name: e3^e6
37: Value: 1.00000E+00 Grade: 3 Base: 1 0 1 0 0 1 0 0 Name: e1^e3^e6
38: Value: 1.00000E+00 Grade: 3 Base: 0 1 1 0 0 1 0 0 Name: e2^e3^e6
39: Value: 1.00000E+00 Grade: 4 Base: 1 1 1 0 0 1 0 0 Name: e1^e2^e3^e6
40: Value: 1.00000E+00 Grade: 2 Base: 0 0 0 1 0 1 0 0 Name: e4^e6
41: Value: 1.00000E+00 Grade: 3 Base: 1 0 0 1 0 1 0 0 Name: e1^e4^e6
42: Value: 1.00000E+00 Grade: 3 Base: 0 1 0 1 0 1 0 0 Name: e2^e4^e6
43: Value: 1.00000E+00 Grade: 4 Base: 1 1 0 1 0 1 0 0 Name: e1^e2^e4^e6
44: Value: 1.00000E+00 Grade: 3 Base: 0 0 1 1 0 1 0 0 Name: e3^e4^e6
45: Value: 1.00000E+00 Grade: 4 Base: 1 0 1 1 0 1 0 0 Name: e1^e3^e4^e6
46: Value: 1.00000E+00 Grade: 4 Base: 0 1 1 1 0 1 0 0 Name: e2^e3^e4^e6
47: Value: 1.00000E+00 Grade: 5 Base: 1 1 1 1 0 1 0 0 Name: e1^e2^e3^e4^e6
48: Value: 1.00000E+00 Grade: 2 Base: 0 0 0 0 1 1 0 0 Name: e5^e6
49: Value: 1.00000E+00 Grade: 3 Base: 1 0 0 0 1 1 0 0 Name: e1^e5^e6
50: Value: 1.00000E+00 Grade: 3 Base: 0 1 0 0 1 1 0 0 Name: e2^e5^e6
51: Value: 1.00000E+00 Grade: 4 Base: 1 1 0 0 1 1 0 0 Name: e1^e2^e5^e6
52: Value: 1.00000E+00 Grade: 3 Base: 0 0 1 0 1 1 0 0 Name: e3^e5^e6
53: Value: 1.00000E+00 Grade: 4 Base: 1 0 1 0 1 1 0 0 Name: e1^e3^e5^e6
54: Value: 1.00000E+00 Grade: 4 Base: 0 1 1 0 1 1 0 0 Name: e2^e3^e5^e6
55: Value: 1.00000E+00 Grade: 5 Base: 1 1 1 0 1 1 0 0 Name: e1^e2^e3^e5^e6
56: Value: 1.00000E+00 Grade: 3 Base: 0 0 0 1 1 1 0 0 Name: e4^e5^e6
57: Value: 1.00000E+00 Grade: 4 Base: 1 0 0 1 1 1 0 0 Name: e1^e4^e5^e6
58: Value: 1.00000E+00 Grade: 4 Base: 0 1 0 1 1 1 0 0 Name: e2^e4^e5^e6
59: Value: 1.00000E+00 Grade: 5 Base: 1 1 0 1 1 1 0 0 Name: e1^e2^e4^e5^e6
60: Value: 1.00000E+00 Grade: 4 Base: 0 0 1 1 1 1 0 0 Name: e3^e4^e5^e6
61: Value: 1.00000E+00 Grade: 5 Base: 1 0 1 1 1 1 0 0 Name: e1^e3^e4^e5^e6
62: Value: 1.00000E+00 Grade: 5 Base: 0 1 1 1 1 1 0 0 Name: e2^e3^e4^e5^e6
63: Value: 1.00000E+00 Grade: 6 Base: 1 1 1 1 1 1 0 0 Name: e1^e2^e3^e4^e5^e6
64: Value: 1.00000E+00 Grade: 1 Base: 0 0 0 0 0 0 1 0 Name: e7
65: Value: 1.00000E+00 Grade: 2 Base: 1 0 0 0 0 0 1 0 Name: e1^e7
66: Value: 1.00000E+00 Grade: 2 Base: 0 1 0 0 0 0 1 0 Name: e2^e7
67: Value: 1.00000E+00 Grade: 3 Base: 1 1 0 0 0 0 1 0 Name: e1^e2^e7
68: Value: 1.00000E+00 Grade: 2 Base: 0 0 1 0 0 0 1 0 Name: e3^e7
69: Value: 1.00000E+00 Grade: 3 Base: 1 0 1 0 0 0 1 0 Name: e1^e3^e7
70: Value: 1.00000E+00 Grade: 3 Base: 0 1 1 0 0 0 1 0 Name: e2^e3^e7
71: Value: 1.00000E+00 Grade: 4 Base: 1 1 1 0 0 0 1 0 Name: e1^e2^e3^e7
72: Value: 1.00000E+00 Grade: 2 Base: 0 0 0 1 0 0 1 0 Name: e4^e7
73: Value: 1.00000E+00 Grade: 3 Base: 1 0 0 1 0 0 1 0 Name: e1^e4^e7
74: Value: 1.00000E+00 Grade: 3 Base: 0 1 0 1 0 0 1 0 Name: e2^e4^e7
75: Value: 1.00000E+00 Grade: 4 Base: 1 1 0 1 0 0 1 0 Name: e1^e2^e4^e7
76: Value: 1.00000E+00 Grade: 3 Base: 0 0 1 1 0 0 1 0 Name: e3^e4^e7
77: Value: 1.00000E+00 Grade: 4 Base: 1 0 1 1 0 0 1 0 Name: e1^e3^e4^e7
78: Value: 1.00000E+00 Grade: 4 Base: 0 1 1 1 0 0 1 0 Name: e2^e3^e4^e7
79: Value: 1.00000E+00 Grade: 5 Base: 1 1 1 1 0 0 1 0 Name: e1^e2^e3^e4^e7
80: Value: 1.00000E+00 Grade: 2 Base: 0 0 0 0 1 0 1 0 Name: e5^e7
81: Value: 1.00000E+00 Grade: 3 Base: 1 0 0 0 1 0 1 0 Name: e1^e5^e7
82: Value: 1.00000E+00 Grade: 3 Base: 0 1 0 0 1 0 1 0 Name: e2^e5^e7
83: Value: 1.00000E+00 Grade: 4 Base: 1 1 0 0 1 0 1 0 Name: e1^e2^e5^e7
84: Value: 1.00000E+00 Grade: 3 Base: 0 0 1 0 1 0 1 0 Name: e3^e5^e7
85: Value: 1.00000E+00 Grade: 4 Base: 1 0 1 0 1 0 1 0 Name: e1^e3^e5^e7
86: Value: 1.00000E+00 Grade: 4 Base: 0 1 1 0 1 0 1 0 Name: e2^e3^e5^e7
87: Value: 1.00000E+00 Grade: 5 Base: 1 1 1 0 1 0 1 0 Name: e1^e2^e3^e5^e7
88: Value: 1.00000E+00 Grade: 3 Base: 0 0 0 1 1 0 1 0 Name: e4^e5^e7
89: Value: 1.00000E+00 Grade: 4 Base: 1 0 0 1 1 0 1 0 Name: e1^e4^e5^e7
90: Value: 1.00000E+00 Grade: 4 Base: 0 1 0 1 1 0 1 0 Name: e2^e4^e5^e7
91: Value: 1.00000E+00 Grade: 5 Base: 1 1 0 1 1 0 1 0 Name: e1^e2^e4^e5^e7
92: Value: 1.00000E+00 Grade: 4 Base: 0 0 1 1 1 0 1 0 Name: e3^e4^e5^e7
93: Value: 1.00000E+00 Grade: 5 Base: 1 0 1 1 1 0 1 0 Name: e1^e3^e4^e5^e7
94: Value: 1.00000E+00 Grade: 5 Base: 0 1 1 1 1 0 1 0 Name: e2^e3^e4^e5^e7
95: Value: 1.00000E+00 Grade: 6 Base: 1 1 1 1 1 0 1 0 Name: e1^e2^e3^e4^e5^e7
96: Value: 1.00000E+00 Grade: 2 Base: 0 0 0 0 0 1 1 0 Name: e6^e7
97: Value: 1.00000E+00 Grade: 3 Base: 1 0 0 0 0 1 1 0 Name: e1^e6^e7
98: Value: 1.00000E+00 Grade: 3 Base: 0 1 0 0 0 1 1 0 Name: e2^e6^e7
99: Value: 1.00000E+00 Grade: 4 Base: 1 1 0 0 0 1 1 0 Name: e1^e2^e6^e7
100: Value: 1.00000E+00 Grade: 3 Base: 0 0 1 0 0 1 1 0 Name: e3^e6^e7
101: Value: 1.00000E+00 Grade: 4 Base: 1 0 1 0 0 1 1 0 Name: e1^e3^e6^e7
102: Value: 1.00000E+00 Grade: 4 Base: 0 1 1 0 0 1 1 0 Name: e2^e3^e6^e7
103: Value: 1.00000E+00 Grade: 5 Base: 1 1 1 0 0 1 1 0 Name: e1^e2^e3^e6^e7
104: Value: 1.00000E+00 Grade: 3 Base: 0 0 0 1 0 1 1 0 Name: e4^e6^e7
105: Value: 1.00000E+00 Grade: 4 Base: 1 0 0 1 0 1 1 0 Name: e1^e4^e6^e7
106: Value: 1.00000E+00 Grade: 4 Base: 0 1 0 1 0 1 1 0 Name: e2^e4^e6^e7
107: Value: 1.00000E+00 Grade: 5 Base: 1 1 0 1 0 1 1 0 Name: e1^e2^e4^e6^e7
108: Value: 1.00000E+00 Grade: 4 Base: 0 0 1 1 0 1 1 0 Name: e3^e4^e6^e7
109: Value: 1.00000E+00 Grade: 5 Base: 1 0 1 1 0 1 1 0 Name: e1^e3^e4^e6^e7
110: Value: 1.00000E+00 Grade: 5 Base: 0 1 1 1 0 1 1 0 Name: e2^e3^e4^e6^e7
111: Value: 1.00000E+00 Grade: 6 Base: 1 1 1 1 0 1 1 0 Name: e1^e2^e3^e4^e6^e7
112: Value: 1.00000E+00 Grade: 3 Base: 0 0 0 0 1 1 1 0 Name: e5^e6^e7
113: Value: 1.00000E+00 Grade: 4 Base: 1 0 0 0 1 1 1 0 Name: e1^e5^e6^e7
114: Value: 1.00000E+00 Grade: 4 Base: 0 1 0 0 1 1 1 0 Name: e2^e5^e6^e7
115: Value: 1.00000E+00 Grade: 5 Base: 1 1 0 0 1 1 1 0 Name: e1^e2^e5^e6^e7
116: Value: 1.00000E+00 Grade: 4 Base: 0 0 1 0 1 1 1 0 Name: e3^e5^e6^e7
117: Value: 1.00000E+00 Grade: 5 Base: 1 0 1 0 1 1 1 0 Name: e1^e3^e5^e6^e7
118: Value: 1.00000E+00 Grade: 5 Base: 0 1 1 0 1 1 1 0 Name: e2^e3^e5^e6^e7
119: Value: 1.00000E+00 Grade: 6 Base: 1 1 1 0 1 1 1 0 Name: e1^e2^e3^e5^e6^e7
120: Value: 1.00000E+00 Grade: 4 Base: 0 0 0 1 1 1 1 0 Name: e4^e5^e6^e7
121: Value: 1.00000E+00 Grade: 5 Base: 1 0 0 1 1 1 1 0 Name: e1^e4^e5^e6^e7
122: Value: 1.00000E+00 Grade: 5 Base: 0 1 0 1 1 1 1 0 Name: e2^e4^e5^e6^e7
123: Value: 1.00000E+00 Grade: 6 Base: 1 1 0 1 1 1 1 0 Name: e1^e2^e4^e5^e6^e7
124: Value: 1.00000E+00 Grade: 5 Base: 0 0 1 1 1 1 1 0 Name: e3^e4^e5^e6^e7
125: Value: 1.00000E+00 Grade: 6 Base: 1 0 1 1 1 1 1 0 Name: e1^e3^e4^e5^e6^e7
126: Value: 1.00000E+00 Grade: 6 Base: 0 1 1 1 1 1 1 0 Name: e2^e3^e4^e5^e6^e7
127: Value: 1.00000E+00 Grade: 7 Base: 1 1 1 1 1 1 1 0 Name: e1^e2^e3^e4^e5^e6^e7
128: Value: 1.00000E+00 Grade: 1 Base: 0 0 0 0 0 0 0 1 Name: e8
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130: Value: 1.00000E+00 Grade: 2 Base: 0 1 0 0 0 0 0 1 Name: e2^e8
131: Value: 1.00000E+00 Grade: 3 Base: 1 1 0 0 0 0 0 1 Name: e1^e2^e8
132: Value: 1.00000E+00 Grade: 2 Base: 0 0 1 0 0 0 0 1 Name: e3^e8
133: Value: 1.00000E+00 Grade: 3 Base: 1 0 1 0 0 0 0 1 Name: e1^e3^e8
134: Value: 1.00000E+00 Grade: 3 Base: 0 1 1 0 0 0 0 1 Name: e2^e3^e8
135: Value: 1.00000E+00 Grade: 4 Base: 1 1 1 0 0 0 0 1 Name: e1^e2^e3^e8
136: Value: 1.00000E+00 Grade: 2 Base: 0 0 0 1 0 0 0 1 Name: e4^e8
137: Value: 1.00000E+00 Grade: 3 Base: 1 0 0 1 0 0 0 1 Name: e1^e4^e8
138: Value: 1.00000E+00 Grade: 3 Base: 0 1 0 1 0 0 0 1 Name: e2^e4^e8
139: Value: 1.00000E+00 Grade: 4 Base: 1 1 0 1 0 0 0 1 Name: e1^e2^e4^e8
140: Value: 1.00000E+00 Grade: 3 Base: 0 0 1 1 0 0 0 1 Name: e3^e4^e8
141: Value: 1.00000E+00 Grade: 4 Base: 1 0 1 1 0 0 0 1 Name: e1^e3^e4^e8
142: Value: 1.00000E+00 Grade: 4 Base: 0 1 1 1 0 0 0 1 Name: e2^e3^e4^e8
143: Value: 1.00000E+00 Grade: 5 Base: 1 1 1 1 0 0 0 1 Name: e1^e2^e3^e4^e8
144: Value: 1.00000E+00 Grade: 2 Base: 0 0 0 0 1 0 0 1 Name: e5^e8
145: Value: 1.00000E+00 Grade: 3 Base: 1 0 0 0 1 0 0 1 Name: e1^e5^e8
146: Value: 1.00000E+00 Grade: 3 Base: 0 1 0 0 1 0 0 1 Name: e2^e5^e8
147: Value: 1.00000E+00 Grade: 4 Base: 1 1 0 0 1 0 0 1 Name: e1^e2^e5^e8
148: Value: 1.00000E+00 Grade: 3 Base: 0 0 1 0 1 0 0 1 Name: e3^e5^e8
149: Value: 1.00000E+00 Grade: 4 Base: 1 0 1 0 1 0 0 1 Name: e1^e3^e5^e8
150: Value: 1.00000E+00 Grade: 4 Base: 0 1 1 0 1 0 0 1 Name: e2^e3^e5^e8
151: Value: 1.00000E+00 Grade: 5 Base: 1 1 1 0 1 0 0 1 Name: e1^e2^e3^e5^e8
152: Value: 1.00000E+00 Grade: 3 Base: 0 0 0 1 1 0 0 1 Name: e4^e5^e8
153: Value: 1.00000E+00 Grade: 4 Base: 1 0 0 1 1 0 0 1 Name: e1^e4^e5^e8
154: Value: 1.00000E+00 Grade: 4 Base: 0 1 0 1 1 0 0 1 Name: e2^e4^e5^e8
155: Value: 1.00000E+00 Grade: 5 Base: 1 1 0 1 1 0 0 1 Name: e1^e2^e4^e5^e8
156: Value: 1.00000E+00 Grade: 4 Base: 0 0 1 1 1 0 0 1 Name: e3^e4^e5^e8
157: Value: 1.00000E+00 Grade: 5 Base: 1 0 1 1 1 0 0 1 Name: e1^e3^e4^e5^e8
158: Value: 1.00000E+00 Grade: 5 Base: 0 1 1 1 1 0 0 1 Name: e2^e3^e4^e5^e8
159: Value: 1.00000E+00 Grade: 6 Base: 1 1 1 1 1 0 0 1 Name: e1^e2^e3^e4^e5^e8
160: Value: 1.00000E+00 Grade: 2 Base: 0 0 0 0 0 1 0 1 Name: e6^e8
161: Value: 1.00000E+00 Grade: 3 Base: 1 0 0 0 0 1 0 1 Name: e1^e6^e8
162: Value: 1.00000E+00 Grade: 3 Base: 0 1 0 0 0 1 0 1 Name: e2^e6^e8
163: Value: 1.00000E+00 Grade: 4 Base: 1 1 0 0 0 1 0 1 Name: e1^e2^e6^e8
164: Value: 1.00000E+00 Grade: 3 Base: 0 0 1 0 0 1 0 1 Name: e3^e6^e8
165: Value: 1.00000E+00 Grade: 4 Base: 1 0 1 0 0 1 0 1 Name: e1^e3^e6^e8
166: Value: 1.00000E+00 Grade: 4 Base: 0 1 1 0 0 1 0 1 Name: e2^e3^e6^e8
167: Value: 1.00000E+00 Grade: 5 Base: 1 1 1 0 0 1 0 1 Name: e1^e2^e3^e6^e8
168: Value: 1.00000E+00 Grade: 3 Base: 0 0 0 1 0 1 0 1 Name: e4^e6^e8
169: Value: 1.00000E+00 Grade: 4 Base: 1 0 0 1 0 1 0 1 Name: e1^e4^e6^e8
170: Value: 1.00000E+00 Grade: 4 Base: 0 1 0 1 0 1 0 1 Name: e2^e4^e6^e8
171: Value: 1.00000E+00 Grade: 5 Base: 1 1 0 1 0 1 0 1 Name: e1^e2^e4^e6^e8
172: Value: 1.00000E+00 Grade: 4 Base: 0 0 1 1 0 1 0 1 Name: e3^e4^e6^e8
173: Value: 1.00000E+00 Grade: 5 Base: 1 0 1 1 0 1 0 1 Name: e1^e3^e4^e6^e8
174: Value: 1.00000E+00 Grade: 5 Base: 0 1 1 1 0 1 0 1 Name: e2^e3^e4^e6^e8
175: Value: 1.00000E+00 Grade: 6 Base: 1 1 1 1 0 1 0 1 Name: e1^e2^e3^e4^e6^e8
176: Value: 1.00000E+00 Grade: 3 Base: 0 0 0 0 1 1 0 1 Name: e5^e6^e8
177: Value: 1.00000E+00 Grade: 4 Base: 1 0 0 0 1 1 0 1 Name: e1^e5^e6^e8
178: Value: 1.00000E+00 Grade: 4 Base: 0 1 0 0 1 1 0 1 Name: e2^e5^e6^e8
179: Value: 1.00000E+00 Grade: 5 Base: 1 1 0 0 1 1 0 1 Name: e1^e2^e5^e6^e8
180: Value: 1.00000E+00 Grade: 4 Base: 0 0 1 0 1 1 0 1 Name: e3^e5^e6^e8
181: Value: 1.00000E+00 Grade: 5 Base: 1 0 1 0 1 1 0 1 Name: e1^e3^e5^e6^e8
182: Value: 1.00000E+00 Grade: 5 Base: 0 1 1 0 1 1 0 1 Name: e2^e3^e5^e6^e8
183: Value: 1.00000E+00 Grade: 6 Base: 1 1 1 0 1 1 0 1 Name: e1^e2^e3^e5^e6^e8
184: Value: 1.00000E+00 Grade: 4 Base: 0 0 0 1 1 1 0 1 Name: e4^e5^e6^e8
185: Value: 1.00000E+00 Grade: 5 Base: 1 0 0 1 1 1 0 1 Name: e1^e4^e5^e6^e8
186: Value: 1.00000E+00 Grade: 5 Base: 0 1 0 1 1 1 0 1 Name: e2^e4^e5^e6^e8
187: Value: 1.00000E+00 Grade: 6 Base: 1 1 0 1 1 1 0 1 Name: e1^e2^e4^e5^e6^e8
188: Value: 1.00000E+00 Grade: 5 Base: 0 0 1 1 1 1 0 1 Name: e3^e4^e5^e6^e8
189: Value: 1.00000E+00 Grade: 6 Base: 1 0 1 1 1 1 0 1 Name: e1^e3^e4^e5^e6^e8
190: Value: 1.00000E+00 Grade: 6 Base: 0 1 1 1 1 1 0 1 Name: e2^e3^e4^e5^e6^e8
191: Value: 1.00000E+00 Grade: 7 Base: 1 1 1 1 1 1 0 1 Name: e1^e2^e3^e4^e5^e6^e8
192: Value: 1.00000E+00 Grade: 2 Base: 0 0 0 0 0 0 1 1 Name: e7^e8
193: Value: 1.00000E+00 Grade: 3 Base: 1 0 0 0 0 0 1 1 Name: e1^e7^e8
194: Value: 1.00000E+00 Grade: 3 Base: 0 1 0 0 0 0 1 1 Name: e2^e7^e8
195: Value: 1.00000E+00 Grade: 4 Base: 1 1 0 0 0 0 1 1 Name: e1^e2^e7^e8
196: Value: 1.00000E+00 Grade: 3 Base: 0 0 1 0 0 0 1 1 Name: e3^e7^e8
197: Value: 1.00000E+00 Grade: 4 Base: 1 0 1 0 0 0 1 1 Name: e1^e3^e7^e8
198: Value: 1.00000E+00 Grade: 4 Base: 0 1 1 0 0 0 1 1 Name: e2^e3^e7^e8
199: Value: 1.00000E+00 Grade: 5 Base: 1 1 1 0 0 0 1 1 Name: e1^e2^e3^e7^e8
200: Value: 1.00000E+00 Grade: 3 Base: 0 0 0 1 0 0 1 1 Name: e4^e7^e8
201: Value: 1.00000E+00 Grade: 4 Base: 1 0 0 1 0 0 1 1 Name: e1^e4^e7^e8
202: Value: 1.00000E+00 Grade: 4 Base: 0 1 0 1 0 0 1 1 Name: e2^e4^e7^e8
203: Value: 1.00000E+00 Grade: 5 Base: 1 1 0 1 0 0 1 1 Name: e1^e2^e4^e7^e8
204: Value: 1.00000E+00 Grade: 4 Base: 0 0 1 1 0 0 1 1 Name: e3^e4^e7^e8
205: Value: 1.00000E+00 Grade: 5 Base: 1 0 1 1 0 0 1 1 Name: e1^e3^e4^e7^e8
206: Value: 1.00000E+00 Grade: 5 Base: 0 1 1 1 0 0 1 1 Name: e2^e3^e4^e7^e8
207: Value: 1.00000E+00 Grade: 6 Base: 1 1 1 1 0 0 1 1 Name: e1^e2^e3^e4^e7^e8
208: Value: 1.00000E+00 Grade: 3 Base: 0 0 0 0 1 0 1 1 Name: e5^e7^e8
209: Value: 1.00000E+00 Grade: 4 Base: 1 0 0 0 1 0 1 1 Name: e1^e5^e7^e8
210: Value: 1.00000E+00 Grade: 4 Base: 0 1 0 0 1 0 1 1 Name: e2^e5^e7^e8
211: Value: 1.00000E+00 Grade: 5 Base: 1 1 0 0 1 0 1 1 Name: e1^e2^e5^e7^e8
212: Value: 1.00000E+00 Grade: 4 Base: 0 0 1 0 1 0 1 1 Name: e3^e5^e7^e8
213: Value: 1.00000E+00 Grade: 5 Base: 1 0 1 0 1 0 1 1 Name: e1^e3^e5^e7^e8
214: Value: 1.00000E+00 Grade: 5 Base: 0 1 1 0 1 0 1 1 Name: e2^e3^e5^e7^e8
215: Value: 1.00000E+00 Grade: 6 Base: 1 1 1 0 1 0 1 1 Name: e1^e2^e3^e5^e7^e8
216: Value: 1.00000E+00 Grade: 4 Base: 0 0 0 1 1 0 1 1 Name: e4^e5^e7^e8
217: Value: 1.00000E+00 Grade: 5 Base: 1 0 0 1 1 0 1 1 Name: e1^e4^e5^e7^e8
218: Value: 1.00000E+00 Grade: 5 Base: 0 1 0 1 1 0 1 1 Name: e2^e4^e5^e7^e8
219: Value: 1.00000E+00 Grade: 6 Base: 1 1 0 1 1 0 1 1 Name: e1^e2^e4^e5^e7^e8
220: Value: 1.00000E+00 Grade: 5 Base: 0 0 1 1 1 0 1 1 Name: e3^e4^e5^e7^e8
221: Value: 1.00000E+00 Grade: 6 Base: 1 0 1 1 1 0 1 1 Name: e1^e3^e4^e5^e7^e8
222: Value: 1.00000E+00 Grade: 6 Base: 0 1 1 1 1 0 1 1 Name: e2^e3^e4^e5^e7^e8
223: Value: 1.00000E+00 Grade: 7 Base: 1 1 1 1 1 0 1 1 Name: e1^e2^e3^e4^e5^e7^e8
224: Value: 1.00000E+00 Grade: 3 Base: 0 0 0 0 0 1 1 1 Name: e6^e7^e8
225: Value: 1.00000E+00 Grade: 4 Base: 1 0 0 0 0 1 1 1 Name: e1^e6^e7^e8
226: Value: 1.00000E+00 Grade: 4 Base: 0 1 0 0 0 1 1 1 Name: e2^e6^e7^e8
227: Value: 1.00000E+00 Grade: 5 Base: 1 1 0 0 0 1 1 1 Name: e1^e2^e6^e7^e8
228: Value: 1.00000E+00 Grade: 4 Base: 0 0 1 0 0 1 1 1 Name: e3^e6^e7^e8
229: Value: 1.00000E+00 Grade: 5 Base: 1 0 1 0 0 1 1 1 Name: e1^e3^e6^e7^e8
230: Value: 1.00000E+00 Grade: 5 Base: 0 1 1 0 0 1 1 1 Name: e2^e3^e6^e7^e8
231: Value: 1.00000E+00 Grade: 6 Base: 1 1 1 0 0 1 1 1 Name: e1^e2^e3^e6^e7^e8
232: Value: 1.00000E+00 Grade: 4 Base: 0 0 0 1 0 1 1 1 Name: e4^e6^e7^e8
233: Value: 1.00000E+00 Grade: 5 Base: 1 0 0 1 0 1 1 1 Name: e1^e4^e6^e7^e8
234: Value: 1.00000E+00 Grade: 5 Base: 0 1 0 1 0 1 1 1 Name: e2^e4^e6^e7^e8
235: Value: 1.00000E+00 Grade: 6 Base: 1 1 0 1 0 1 1 1 Name: e1^e2^e4^e6^e7^e8
236: Value: 1.00000E+00 Grade: 5 Base: 0 0 1 1 0 1 1 1 Name: e3^e4^e6^e7^e8
237: Value: 1.00000E+00 Grade: 6 Base: 1 0 1 1 0 1 1 1 Name: e1^e3^e4^e6^e7^e8
238: Value: 1.00000E+00 Grade: 6 Base: 0 1 1 1 0 1 1 1 Name: e2^e3^e4^e6^e7^e8
239: Value: 1.00000E+00 Grade: 7 Base: 1 1 1 1 0 1 1 1 Name: e1^e2^e3^e4^e6^e7^e8
240: Value: 1.00000E+00 Grade: 4 Base: 0 0 0 0 1 1 1 1 Name: e5^e6^e7^e8
241: Value: 1.00000E+00 Grade: 5 Base: 1 0 0 0 1 1 1 1 Name: e1^e5^e6^e7^e8
242: Value: 1.00000E+00 Grade: 5 Base: 0 1 0 0 1 1 1 1 Name: e2^e5^e6^e7^e8
243: Value: 1.00000E+00 Grade: 6 Base: 1 1 0 0 1 1 1 1 Name: e1^e2^e5^e6^e7^e8
244: Value: 1.00000E+00 Grade: 5 Base: 0 0 1 0 1 1 1 1 Name: e3^e5^e6^e7^e8
245: Value: 1.00000E+00 Grade: 6 Base: 1 0 1 0 1 1 1 1 Name: e1^e3^e5^e6^e7^e8
246: Value: 1.00000E+00 Grade: 6 Base: 0 1 1 0 1 1 1 1 Name: e2^e3^e5^e6^e7^e8
247: Value: 1.00000E+00 Grade: 7 Base: 1 1 1 0 1 1 1 1 Name: e1^e2^e3^e5^e6^e7^e8
248: Value: 1.00000E+00 Grade: 5 Base: 0 0 0 1 1 1 1 1 Name: e4^e5^e6^e7^e8
249: Value: 1.00000E+00 Grade: 6 Base: 1 0 0 1 1 1 1 1 Name: e1^e4^e5^e6^e7^e8
250: Value: 1.00000E+00 Grade: 6 Base: 0 1 0 1 1 1 1 1 Name: e2^e4^e5^e6^e7^e8
251: Value: 1.00000E+00 Grade: 7 Base: 1 1 0 1 1 1 1 1 Name: e1^e2^e4^e5^e6^e7^e8
252: Value: 1.00000E+00 Grade: 6 Base: 0 0 1 1 1 1 1 1 Name: e3^e4^e5^e6^e7^e8
253: Value: 1.00000E+00 Grade: 7 Base: 1 0 1 1 1 1 1 1 Name: e1^e3^e4^e5^e6^e7^e8
254: Value: 1.00000E+00 Grade: 7 Base: 0 1 1 1 1 1 1 1 Name: e2^e3^e4^e5^e6^e7^e8
255: Value: 1.00000E+00 Grade: 8 Base: 1 1 1 1 1 1 1 1 Name: e1^e2^e3^e4^e5^e6^e7^e8
Title: Products
Post by: hermann on September 06, 2014, 08:58:31 AM
Here are some multiplications tables for products of multivectors:
2-Dimensions:
3-Dimensions
2-Dimensions:
3-Dimensions
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on September 06, 2014, 09:11:37 AM
4-Dimensions:
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 06, 2014, 10:55:30 AM
since the first row and line are trivially the same as the second row and line, I like to leave them out in multiplication tables.
These are some interesting patterns, there. I'd love to see them colorcoded or something. It's hard for the eye to tell anything interesting.
Due to the binary formulation, this is sort of "counting with subspaces" :D
These are some interesting patterns, there. I'd love to see them colorcoded or something. It's hard for the eye to tell anything interesting.
Due to the binary formulation, this is sort of "counting with subspaces" :D
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on September 06, 2014, 02:30:44 PM
I original had the first row and the first coloumn of the table in a different color.
But the LaTex here doesen't accept the following statements:
\rowcolor{cyan}
\cellcolor{cyan}
may be the
\usepackage{colortbl}
can be inserted in the LaTex installation here in fractal forums.
Hermann
But the LaTex here doesen't accept the following statements:
\rowcolor{cyan}
\cellcolor{cyan}
may be the
\usepackage{colortbl}
can be inserted in the LaTex installation here in fractal forums.
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 06, 2014, 03:07:58 PM
but you don't even need those at all, since they are exact copies of the second column/line (since any thing * 1 = thing)
They just clutter the tables a bit and make them larger than necessary.
They just clutter the tables a bit and make them larger than necessary.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on September 06, 2014, 03:44:13 PM
Also for readability I find the e12 shorthand easier than explicit e1^e2.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 06, 2014, 04:21:37 PM
Very agreed
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on September 07, 2014, 09:00:58 AM
Not a big problem.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 07, 2014, 12:13:44 PM
much better, thanks. :)
Now the only thing that's missing is color to distinguish them all properly :D
Are you already able to create negative rules too?
If you want to go to a conformal geometric algebra, or special relativity, for instance, you'll need a 1-vector element that squares to -1.
I also had this idea for an "infinite-dimensional" version going in both directions and wonder if the math within that checks out (I know that it is possible in principle: it was referenced before, but I never saw an example of it)
Basically, you'd have
with:
Obviously, for something like this, you can't ever generate the full multiplication table, and you gotta do calculations on the fly, but as long as the whole thing is sparse, e.g. only finitely many of the values are used, you should be able to use a model like this just fine.
Now the only thing that's missing is color to distinguish them all properly :D
Are you already able to create negative rules too?
If you want to go to a conformal geometric algebra, or special relativity, for instance, you'll need a 1-vector element that squares to -1.
I also had this idea for an "infinite-dimensional" version going in both directions and wonder if the math within that checks out (I know that it is possible in principle: it was referenced before, but I never saw an example of it)
Basically, you'd have
Obviously, for something like this, you can't ever generate the full multiplication table, and you gotta do calculations on the fly, but as long as the whole thing is sparse, e.g. only finitely many of the values are used, you should be able to use a model like this just fine.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on September 07, 2014, 03:08:42 PM
Thanks,
Some Background Information. I tested the generated LaTex table in a LaTex-Document first.
I gave the first row and the first column the background color cyan.
That worked fine on my computer.
When I then posted the LaTex here in Fractalfoums, I got the problem that the color commands, for table items, are not accepted! See my post above!
It also interpretated the whole table as formula, so that I had to get rid of $ symbols surrunding each item.
I also started to work on an HTML-Table generator. With this approce it is easier to change the background color of the lines, or the items of a table.
At the moment my internet connection is slow, so I couldn't load it up. On my computer I was able to create tables up to 8-Dimensions but got storage problems for higher dimensions.
Signature: I have not implemented it but it should not be too difficult to program.
Dimension : Natural := 3;
type Signature_Type is array (0 .. Dimension) of Integer with Default_Component_Value => 1;
Signature : Signature_Type := (1, -1, 0, 1);
At the moment the dimension value can start with 0; (even the implementaion works at the moment only with values >1)
May be it is possible to expand it to negative Dimensions:
type Signature_Type is array ( -Dimension .. Dimension ) of Integer with Default_Component_Value => 1;
Signature : Signature_Type := (2,-3-0, 5, -1, 0, 1);
Set_Signature (Signature);
It is also possible to think of float values as elements of the signature.
Hermann
Some Background Information. I tested the generated LaTex table in a LaTex-Document first.
I gave the first row and the first column the background color cyan.
That worked fine on my computer.
When I then posted the LaTex here in Fractalfoums, I got the problem that the color commands, for table items, are not accepted! See my post above!
It also interpretated the whole table as formula, so that I had to get rid of $ symbols surrunding each item.
I also started to work on an HTML-Table generator. With this approce it is easier to change the background color of the lines, or the items of a table.
At the moment my internet connection is slow, so I couldn't load it up. On my computer I was able to create tables up to 8-Dimensions but got storage problems for higher dimensions.
Signature: I have not implemented it but it should not be too difficult to program.
Dimension : Natural := 3;
type Signature_Type is array (0 .. Dimension) of Integer with Default_Component_Value => 1;
Signature : Signature_Type := (1, -1, 0, 1);
At the moment the dimension value can start with 0; (even the implementaion works at the moment only with values >1)
May be it is possible to expand it to negative Dimensions:
type Signature_Type is array ( -Dimension .. Dimension ) of Integer with Default_Component_Value => 1;
Signature : Signature_Type := (2,-3-0, 5, -1, 0, 1);
Set_Signature (Signature);
It is also possible to think of float values as elements of the signature.
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 07, 2014, 08:31:28 PM
Heh, no worries about the color, I saw your first remark already. I didn't mean to push you with that repeated remark or anything.
I'd imagine that, at least if it is done like I suggested, this binary-type implementation would kinda break down:
For a short (a single byte), you'd have your eight bits:
_ _ _ _ _ _ _ _
1 in this would be something like
0 0 0 0 0 0 0 1
-1, meanwhile, using the implicit 1, would look like
1 0 0 0 0 0 0 0
and -2 would be
1 0 0 0 0 0 0 1
and it could build numbers from 127 to -128. (or alternatively, you could get a signed 0 where 0 0 0 0 0 0 0 0 = +0 and 1 0 0 0 0 0 0 0 = -0)
More than plenty, except for two problems:
- I am asking for infinitely many dimensions of which, however, only finitely many are used, so this basic format will most definitely fail (gotta use a sparse representation instead)
- since two different numbers use at least one bit twice, you'd have to somehow differentiate between the two extra. As long as you do not multiply any together, it's no problem, but what would you do with
? This obviously can't quite work.
Though that kind of concept, at least for now, goes way past the scope of your implementation here anyway. Something like that could be tried later down the line.
Btw, note that this doesn't mean that there are negative dimensions. It's
many dimensions. And while some of them would get a "negative name", all of them would still behave like positive dimensions.
An alternative approach could be to only use natural numbers, and have even dimension be negativeand odd ones positive. Then you'd have:
etc.
That would have the advantage, on one hand, that you could implement this with your binary representation just fine, but on the other, it'd still have the problem of being infinite in size and thus, ultimately, requiring sparse represenation instead.
There is also the issue of not having an idempotent / pseudoscalar in such an infinite dimensional case. At least not a well-definable one.
For instance, in 2 dimensions, you have
1 x y xy=I
Ix = xyx = -y
xI = xxy = y
Iy = xyy = x
yI = yxy = -x
so I anticommutes with x and y.
Meanwhile, in 3D, you have
x y z and xyz = I
Ix = xyzx = - xyxz = xxyz = yz
xI = xxyz = yz
Iy = xyzy = - xyyz = -xz = zx
yI = yxyz = - yyxz = -xz = zx
Iz = xyzz = xy
zI = zxyz = -xzyz = xyzz = yx
Here, I commutes.
This pattern continues for higher dimensions depending on whether I is an even- or odd-dimensioned space.
In case of an infinite dimensional algebra, it's undefined whether it is even or odd.
If that were no problem, it would be easy to define things like a join and a meet in this space, even with a sparse representation. A sparse dual would simply be the opposite of a sparse vector. Instead of a limited number of 1s, there would be a limited number of 0s. So even there, multiplication should not be a problem.
But because of this even/odd problem, it's not as straight-forward to define this.
Of course, for any finite-dimensional subspace, you could define any given operation just fine.
I'd imagine that, at least if it is done like I suggested, this binary-type implementation would kinda break down:
For a short (a single byte), you'd have your eight bits:
_ _ _ _ _ _ _ _
1 in this would be something like
0 0 0 0 0 0 0 1
-1, meanwhile, using the implicit 1, would look like
1 0 0 0 0 0 0 0
and -2 would be
1 0 0 0 0 0 0 1
and it could build numbers from 127 to -128. (or alternatively, you could get a signed 0 where 0 0 0 0 0 0 0 0 = +0 and 1 0 0 0 0 0 0 0 = -0)
More than plenty, except for two problems:
- I am asking for infinitely many dimensions of which, however, only finitely many are used, so this basic format will most definitely fail (gotta use a sparse representation instead)
- since two different numbers use at least one bit twice, you'd have to somehow differentiate between the two extra. As long as you do not multiply any together, it's no problem, but what would you do with
Though that kind of concept, at least for now, goes way past the scope of your implementation here anyway. Something like that could be tried later down the line.
Btw, note that this doesn't mean that there are negative dimensions. It's
An alternative approach could be to only use natural numbers, and have even dimension be negativeand odd ones positive. Then you'd have:
etc.
That would have the advantage, on one hand, that you could implement this with your binary representation just fine, but on the other, it'd still have the problem of being infinite in size and thus, ultimately, requiring sparse represenation instead.
There is also the issue of not having an idempotent / pseudoscalar in such an infinite dimensional case. At least not a well-definable one.
For instance, in 2 dimensions, you have
1 x y xy=I
Ix = xyx = -y
xI = xxy = y
Iy = xyy = x
yI = yxy = -x
so I anticommutes with x and y.
Meanwhile, in 3D, you have
x y z and xyz = I
Ix = xyzx = - xyxz = xxyz = yz
xI = xxyz = yz
Iy = xyzy = - xyyz = -xz = zx
yI = yxyz = - yyxz = -xz = zx
Iz = xyzz = xy
zI = zxyz = -xzyz = xyzz = yx
Here, I commutes.
This pattern continues for higher dimensions depending on whether I is an even- or odd-dimensioned space.
In case of an infinite dimensional algebra, it's undefined whether it is even or odd.
If that were no problem, it would be easy to define things like a join and a meet in this space, even with a sparse representation. A sparse dual would simply be the opposite of a sparse vector. Instead of a limited number of 1s, there would be a limited number of 0s. So even there, multiplication should not be a problem.
But because of this even/odd problem, it's not as straight-forward to define this.
Of course, for any finite-dimensional subspace, you could define any given operation just fine.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on September 08, 2014, 07:08:26 AM
Hallo Kram,
the scalar product in euclidian metric is usally defined in the following way:
then we have for the outer product squared:
^2 =(e_1 \wedge e_2)(e_1 \wedge e_2) = -(e_1 \wedge e_2)(e_2 \wedge e_1) = -(e_1)1(e_1)= -1)
If we redefine the inner product in the following way:
Handling the outer product as before.
we can set
)
)
)
)
)
and get different inner products.
If we have)
^2 =(e_1 \wedge e_2)(e_1 \wedge e_2) = -(e_1 \wedge e_2)(e_2 \wedge e_1) = -(e_1)-1(e_1)= +1)
If we have)
^2 =(e_1 \wedge e_2)(e_1 \wedge e_2) = -(e_1 \wedge e_2)(e_2 \wedge e_1) = -(e_1)-3.1415(e_1)= +3.1415)
Or a bit more general:
(e_2 \wedge e_3) = m_2(e_1 \wedge e_3))
This shouldn't be too difficult to be implemented. Or did I miss your point?
the scalar product in euclidian metric is usally defined in the following way:
then we have for the outer product squared:
If we redefine the inner product in the following way:
Handling the outer product as before.
we can set
and get different inner products.
If we have
If we have
Or a bit more general:
This shouldn't be too difficult to be implemented. Or did I miss your point?
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 08, 2014, 02:14:06 PM
That's to implement the grade, which is fine, however I was talking about other things too:
- Implementing negative dimensions with your binary representation (it won't work directly, but a remapping is possible)
- Implementing infinitely many dimensions with using only finitely many of them at any given time (having a direct binary representation will fail in this case: you just can't store infinitely many zeroes. However, you can implicitly assume them to be 0 and then directly encode just the placements of the 1s instead - that's a sparse representation)
- calculating things like dual spaces with an infinite dimensional pseudoscalar/idempotent. (I thought further about this and I think I solved it, unless I made an error)
For this, I am assuming
to ONLY be the positive integers, excluding 0. Normally, I would include 0, but the notation becomes nicer if I do this here. (Alternatively you could start counting at 0 with your dimensions, making the first 1-vector 
rather than 
)
Let's define the idempotent of such an infinite-dmensional space as
and the signature of our space as a vector 
with 
.
Doing something like
, for instance, is fairly simple. You'll have to reorder 356 (=357-1) times, which is an even number, so you get  \: m_{357} e_{\left{\mathbb{N }\setminus \left{357 \right} \right})
which can be precisely defined by a sparse representation (namely by exactly inverting the previous sparse representation: you assume all the values except a few to be 1, and then you precisely specify which ones are the 0s)
However, if you go for
, how often do you need to reorder? It depends on the parity of 
:
If your particular infinity is assumed to be even,
If it is odd instead, then
So in the end, you effectively have two different infinitely-dimensional spaces, one for an "even infinity"
and one for an "odd infinity" 
.
- Implementing negative dimensions with your binary representation (it won't work directly, but a remapping is possible)
- Implementing infinitely many dimensions with using only finitely many of them at any given time (having a direct binary representation will fail in this case: you just can't store infinitely many zeroes. However, you can implicitly assume them to be 0 and then directly encode just the placements of the 1s instead - that's a sparse representation)
- calculating things like dual spaces with an infinite dimensional pseudoscalar/idempotent. (I thought further about this and I think I solved it, unless I made an error)
For this, I am assuming
Let's define the idempotent of such an infinite-dmensional space as
Doing something like
However, if you go for
If your particular infinity is assumed to be even,
If it is odd instead, then
So in the end, you effectively have two different infinitely-dimensional spaces, one for an "even infinity"
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on September 08, 2014, 09:16:53 PM
Thanks for your explanation.
In the mean time I have worked a bit on the Software and are now able to create multiplikation tables up to 9 Dimensions in HTML-Format.
It is possible to view this table with the mozilla firefox browser, but the browser becomes slow.
What i didn't expect that I come so soon on the limits of my computer system when working on this project.
Hermann
P.S I am also preparing a Page on my home page to be able to present some of the tables.
In the mean time I have worked a bit on the Software and are now able to create multiplikation tables up to 9 Dimensions in HTML-Format.
It is possible to view this table with the mozilla firefox browser, but the browser becomes slow.
What i didn't expect that I come so soon on the limits of my computer system when working on this project.
- First I discovered, that n_over_k showes soon the limitations of the integer implementation.
- Next, the implementation of indizies became tricky
- Then the problem with big LaTex Tables.
- Next the problem of writing a complete string into a file.
- And know I come to the problem of displaying huge HTML-Tables.
- Idears appear faster then I am able to implement of write down
Hermann
P.S I am also preparing a Page on my home page to be able to present some of the tables.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 09, 2014, 10:26:31 PM
Symplectic, orthogonal and linear Lie groups in
Clifford algebra (pdf) (http://arxiv.org/pdf/1409.2452v1.pdf)
Clifford algebra (pdf) (http://arxiv.org/pdf/1409.2452v1.pdf)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on September 11, 2014, 12:26:00 AM
For the presentation of the tables and the collection of idears and links, I have know started my own internet page on geometric algebra.
The tables generated by the program I have written. I hope, I can but more contens on it the next days.
http://www.wackerart.de/mathematik/geometric_algebra.html (http://www.wackerart.de/mathematik/geometric_algebra.html)
Hermann
The tables generated by the program I have written. I hope, I can but more contens on it the next days.
http://www.wackerart.de/mathematik/geometric_algebra.html (http://www.wackerart.de/mathematik/geometric_algebra.html)
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 11, 2014, 12:34:40 AM
Nice page.
What's up with all those images though?
I mean, they are nice to look at, but they split up the rest of the page more than they probably should and they do not appear to be related. It's also a lot of the Ederlake.
What's up with all those images though?
I mean, they are nice to look at, but they split up the rest of the page more than they probably should and they do not appear to be related. It's also a lot of the Ederlake.
Title: Photos
Post by: hermann on September 11, 2014, 10:37:09 AM
Hallo Kram,
basically some photos on my geometric algebra page
http://www.wackerart.de/mathematik/geometric_algebra.html (http://www.wackerart.de/mathematik/geometric_algebra.html)
are made during the same days when I worked on the HTML page and where writing the software for the creation of the tables.
For me it was big fun making the photos, programming a bit and thinking about geometric algebra while walking around lake "Edersee".
So there is a strong emotional relationship (for me) between the photos and the other work.
On the other hand one has the problem how to visualise higher dimensions?
We live in a three dimensional world. One can think of time as a fourth dimension.
Making a film is then a four dimensionl 0bject. Each point in a four dimensional space time universe can have a distingt color. Making another 3-Dimension.
So we can visiualise 7-Dimensions.
To have a picture from higher dimensional objects we can only have different perspectives and projections of this object to visualise it to the human eye.
So I made photos from different positions at different time with different light and different weather from the lake!
Hermann
basically some photos on my geometric algebra page
http://www.wackerart.de/mathematik/geometric_algebra.html (http://www.wackerart.de/mathematik/geometric_algebra.html)
are made during the same days when I worked on the HTML page and where writing the software for the creation of the tables.
For me it was big fun making the photos, programming a bit and thinking about geometric algebra while walking around lake "Edersee".
So there is a strong emotional relationship (for me) between the photos and the other work.
On the other hand one has the problem how to visualise higher dimensions?
We live in a three dimensional world. One can think of time as a fourth dimension.
Making a film is then a four dimensionl 0bject. Each point in a four dimensional space time universe can have a distingt color. Making another 3-Dimension.
So we can visiualise 7-Dimensions.
To have a picture from higher dimensional objects we can only have different perspectives and projections of this object to visualise it to the human eye.
So I made photos from different positions at different time with different light and different weather from the lake!
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on September 11, 2014, 11:11:15 AM
There's dimensions and then there are dimensions. It's an overloaded term so what do you mean? Take 3D. In Clifford/GA you can define a standard model. 1,3,3,1. A vector takes 3 real values to specify so it's 3 dimensional. A full term requires up to 8 real values, so it's an 8 dimensional vector space (vector is another overloaded term). Augment that to a homogeneous model and you now have 3D space embedded in 4D (a vector takes 4 real values). No hardship here since all non-zero scalar multiples represent the same thing, but it's now a 16D vector space. Add other properties to 3D and you expand the vector space, but it's still just plain old 3D with some extra rules (whatever they may be).
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on September 11, 2014, 02:30:46 PM
For me dimension is the number of base vectors of the form 
A multivector is constructed by even more elements. But these elements depend on the these also emply a different dimensionality.
The problem I was talking about is the question of how to visualise all these elements that can be produced by geoemetric algebra.
As an example, for the mandelbrot set we use the complex plane. In geometric algebra we get soon a lot more planes
depending of the number of base vectors.
A multivector is constructed by even more elements. But these elements depend on the
The problem I was talking about is the question of how to visualise all these elements that can be produced by geoemetric algebra.
As an example, for the mandelbrot set we use the complex plane. In geometric algebra we get soon a lot more planes
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on September 11, 2014, 03:08:48 PM
I'm not following. But I babble a little anyway. Remember than complex numbers in the standard model of 2D in GA are the scalar and the bivector parts and can be formulated that way. So some 'z' point is x + 0 e1 +0 e2 + y e12.
If you wanted to play around with variants in some extended model you'd have to move the x and y into the vector and the equation is reformed in the same way as in my quaternion thread. Humm...you could probably do the two other configurations as well...but I don't think that would be of any interest.
Of course I may not be understanding what you mean by visualize...I've been assuming you mean "thinking about the elements" instead of literally "make a picture". If the latter, then it depends on the equation. In many cases you'll have zeros at fixed positions...otherwise you need to play around with projections.
If you wanted to play around with variants in some extended model you'd have to move the x and y into the vector and the equation is reformed in the same way as in my quaternion thread. Humm...you could probably do the two other configurations as well...but I don't think that would be of any interest.
Of course I may not be understanding what you mean by visualize...I've been assuming you mean "thinking about the elements" instead of literally "make a picture". If the latter, then it depends on the equation. In many cases you'll have zeros at fixed positions...otherwise you need to play around with projections.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 11, 2014, 03:13:10 PM
I just had the weirdest idea:
What would happen if you took a movie (2+1+3D) and just applied typical transformations on it?
I've seen switching space and time axes before. That can cause some interesting effects. But what about rotating the color axes into the spacetime axes? (remapped such that there is no color clipping)
I bet that would be pretty trippy.
Datavisualization deals with the issue of dimensional reduction all the time. There are algorithms which filter the data, automatically decide which parts are the "most significant" (e.g. their structure is the most complex/ they contribute the most to the used data) and, from there, produce a low-dimensional image which can be understood and interpreted by us more easily.
It's not perfect, but it's the best we can do.
What would happen if you took a movie (2+1+3D) and just applied typical transformations on it?
I've seen switching space and time axes before. That can cause some interesting effects. But what about rotating the color axes into the spacetime axes? (remapped such that there is no color clipping)
I bet that would be pretty trippy.
Datavisualization deals with the issue of dimensional reduction all the time. There are algorithms which filter the data, automatically decide which parts are the "most significant" (e.g. their structure is the most complex/ they contribute the most to the used data) and, from there, produce a low-dimensional image which can be understood and interpreted by us more easily.
It's not perfect, but it's the best we can do.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on September 11, 2014, 04:24:26 PM
May be I have used the word dimension for my purposes. But geometric algebra is relative new for me, being classicly educated without geometric algebra a
3-Dimensional space was constructed by.
Now in geometric algebra I have named the following structur 3-dimensional. May be not the correct name in mathematics, but that is it what I mean:
The following is a thing I call 4 dimensional
for 5 dimensions my LaTex code is not accepted. But the html-version works:
http://www.wackerart.de/mathematik/geometric_algebra.html (http://www.wackerart.de/mathematik/geometric_algebra.html)
's form a vector space.
form a bivector space.
form a trivector space
one can continue to construc n-vector spaces.
From this Items one can construct a multivector. Which builds a multivector space.
All the vector spaces defined above are of cause vector spaces in the abstract sence!
Each of these vector spaces has of cause his own individual dimensionality.
Hermann
3-Dimensional space was constructed by
Now in geometric algebra I have named the following structur 3-dimensional. May be not the correct name in mathematics, but that is it what I mean:
The following is a thing I call 4 dimensional
for 5 dimensions my LaTex code is not accepted. But the html-version works:
http://www.wackerart.de/mathematik/geometric_algebra.html (http://www.wackerart.de/mathematik/geometric_algebra.html)
one can continue to construc n-vector spaces.
From this Items one can construct a multivector. Which builds a multivector space.
All the vector spaces defined above are of cause vector spaces in the abstract sence!
Each of these vector spaces has of cause his own individual dimensionality.
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 11, 2014, 04:48:34 PM
those objects are the descriptions of an orthonormal system (this is not actually a requirement, you can do non-orthonormal ones too) of all possible subspaces of a given dimensionality, added together to a larger vector space.
In practice you'll rarely need more than just a couple of elements to be presented at the same time.
It's not clear at all what it means to have a scalar AND a directed stretch AND a directed area AND a directed volume all at the same time.
A dimension, in mathematics, is no more than a number. It need not be visualized per se. It's not of great importance.
Intuition is greater in low-dimensional spaces we can actually grasp mentally, but that's precisely the power of geometric algebra: To intuitively extend this to higher-dimensional spaces, making the notion of reflection almost primitive. (It's just a double-multiplication with an element you want to reflect in, and its inversion, e.g. a sandwich operation).
You can't actually visualize those higher-dimensional spaces, but through geometric algebra, the operations on them become more intuitive. The notions of rotation or reflection become really clear.
In practice you'll rarely need more than just a couple of elements to be presented at the same time.
It's not clear at all what it means to have a scalar AND a directed stretch AND a directed area AND a directed volume all at the same time.
A dimension, in mathematics, is no more than a number. It need not be visualized per se. It's not of great importance.
Intuition is greater in low-dimensional spaces we can actually grasp mentally, but that's precisely the power of geometric algebra: To intuitively extend this to higher-dimensional spaces, making the notion of reflection almost primitive. (It's just a double-multiplication with an element you want to reflect in, and its inversion, e.g. a sandwich operation).
You can't actually visualize those higher-dimensional spaces, but through geometric algebra, the operations on them become more intuitive. The notions of rotation or reflection become really clear.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on September 11, 2014, 05:03:14 PM
I just had the weirdest idea:
What would happen if you took a movie (2+1+3D) and just applied typical transformations on it?
I've seen switching space and time axes before. That can cause some interesting effects. But what about rotating the color axes into the spacetime axes? (remapped such that there is no color clipping)
I bet that would be pretty trippy.
May be one day I have a complete version of geometric algebra software. The next step is then how to define fractals and how visualise them. Sounds like a aweful lot of programming.What would happen if you took a movie (2+1+3D) and just applied typical transformations on it?
I've seen switching space and time axes before. That can cause some interesting effects. But what about rotating the color axes into the spacetime axes? (remapped such that there is no color clipping)
I bet that would be pretty trippy.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 11, 2014, 05:44:24 PM
eh, once you have the basic architecture down, defining any kind of geometric process should be easy enough.
In particular, already in this thread we found how to represent the MSet in Geometric Algebra. Applying that to higher dimensions unfortunately (but expectedly) just reproduces the quaternion-style rotated M-Set, but it works nicely for 2D.
What you gotta do is pick a base direction (the direction which will point in the positive real direction of the classical M-Set) and then just multiply it both sides with a (not reversed) arbitrary vector to get the "z²" part. Then, just add another vector c, and you have your usual situation.
Example:
Input vectors:
chosen base direction:
Process:
 e_1 \left(z_1 e_1 + z_2 e_2 \right) + c_1 e_1 + c_2 e_2 = \\ \\<br /><br />\left(z_1 e_1 e_1 + z_2 e_2 e_1 \right) \left(z_1 e_1 + z_2 e_2 \right) + c_1 e_1 + c_2 e_2 = \\ \\<br /><br />z_1^2 e_1 + z^2 z_1 e_2 + z_1 z_2 e_2 + z_2^2 e_2 e_1 e_2 + c_1 e_1 + c_2 e_2 = \\ \\<br /><br />\left(z_1^2 - z_2^2 + c_1 \right) e_1 + \left(2 z_1 z_2 + c_2 \right) e_2 \to z \\\\<br />z_1 \to z_1^2 - z_2^2 +c_1\\<br />z_2 \to 2 z_1 z_2 + c_2 \\<br />z \to z e_1 z + c)
In particular, already in this thread we found how to represent the MSet in Geometric Algebra. Applying that to higher dimensions unfortunately (but expectedly) just reproduces the quaternion-style rotated M-Set, but it works nicely for 2D.
What you gotta do is pick a base direction (the direction which will point in the positive real direction of the classical M-Set) and then just multiply it both sides with a (not reversed) arbitrary vector to get the "z²" part. Then, just add another vector c, and you have your usual situation.
Example:
Input vectors:
chosen base direction:
Process:
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on September 12, 2014, 04:27:14 PM
I am still working on V9 and V18!
Maybe you can check my V9. Attention: This page will challange your browser!
http://www.wackerart.de/mathematik/geometric_algebra/multiplication_table_short_9.html (http://www.wackerart.de/mathematik/geometric_algebra/multiplication_table_short_9.html)
:dink: How did you note V18 with paper and pencil? :dink:
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 12, 2014, 04:38:57 PM
it's not technically hard to write all this down, even if resulting tables are absolutely massive. Though if you multiply in a non-systematic way, by brute-forcing each possible combination of elements one by one to obtain the full multiplication tables, it's a very mind-numbing if technically easy task.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on September 12, 2014, 04:47:32 PM
eh, once you have the basic architecture down, defining any kind of geometric process should be easy enough.
In particular, already in this thread we found how to represent the MSet in Geometric Algebra. Applying that to higher dimensions unfortunately (but expectedly) just reproduces the quaternion-style rotated M-Set, but it works nicely for 2D.
In particular, already in this thread we found how to represent the MSet in Geometric Algebra. Applying that to higher dimensions unfortunately (but expectedly) just reproduces the quaternion-style rotated M-Set, but it works nicely for 2D.
Hallo Kram,
I have no problem in programming M-Sets with geometric algebra. Its more a technical question how to get a graphic interface to my software.
At the moment I am thinking to implement a 3D-JavaScript intefrace on my homepage. With severel windows in each window running a 3D animation.
The only problem is, I have to program it.
Hermann
P.S Something like this: http://www.wackerart.de/windows.html (http://www.wackerart.de/windows.html) (use the scroll bars!)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 12, 2014, 05:12:08 PM
hah that's a neat demo
Title: Canonic Base
Post by: hermann on September 17, 2014, 07:30:26 PM
I have produced the following page with tables of the canonical base vectors for the construction of multi vectors in different dimensions.
http://www.wackerart.de/mathematik/geometric_algebra/several_base_vector_tables.html (http://www.wackerart.de/mathematik/geometric_algebra/several_base_vector_tables.html)
Hermann
http://www.wackerart.de/mathematik/geometric_algebra/several_base_vector_tables.html (http://www.wackerart.de/mathematik/geometric_algebra/several_base_vector_tables.html)
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on September 26, 2014, 11:50:39 PM
With geometic algebra it is possible to build complex numbers and quaternions.
But is it also possible to build Octonions and Sedenions with Geometric Algebra?
They also seems to have very interesting properties.
http://en.wikipedia.org/wiki/Octonion (http://en.wikipedia.org/wiki/Octonion)
http://en.wikipedia.org/wiki/Sedenion (http://en.wikipedia.org/wiki/Sedenion)
Hermann
But is it also possible to build Octonions and Sedenions with Geometric Algebra?
They also seems to have very interesting properties.
http://en.wikipedia.org/wiki/Octonion (http://en.wikipedia.org/wiki/Octonion)
http://en.wikipedia.org/wiki/Sedenion (http://en.wikipedia.org/wiki/Sedenion)
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 27, 2014, 12:50:56 AM
I'm pretty sure yes, however the correspondence might not be quite as pretty.
Quaternions are directly encoded by taking a 3D Clifford algebra together with the scalar element.
Trying to do this with Octonions and Sedenions is not quite as straight forward.
a 6D Clifford algebra happens to have 15 different bivectors, however, this sub-algebra would also generate quadvectors and the hexvector for this space. So you'd end up with 32 distinct elements in that space.
However, a 4D Clifford Algebra has 6 bivectors and its one pseudo-scalar quadvector. This, together with its scalar, should generate Octonions.
And a 5D Clifford Algebra generates 10 bivectors and 5 quadvectors. Taking those plus scalar should be able to generate the Sedenions.
I did not actually test if those are equivalent, but that's the easiest way of how I could see those algebras.
I did test something quickly though: The subalgebra of even-dimensional subspaces of an n-D Clifford Algebra actually has dimension
for any dimension n. So my guess is that all those even-dimensional subspace subalgebras are precisely the, err, -ions.
Note that this trend also fits with the complex numbers. You have your elements {1, x, y, xy=I} of which the even sub-algebra is {1,I} and this happens to precisely model the complex numbers with the dimension
.
This would interestingly mean that, technically, octonions are made up not just of planar rotations (or double-reflections) but also of 4-space rotations (4x-reflections) and sedenions even have a full-space rotation (6 consecutive reflections in different directions) of sorts.
Quaternions are directly encoded by taking a 3D Clifford algebra together with the scalar element.
Trying to do this with Octonions and Sedenions is not quite as straight forward.
a 6D Clifford algebra happens to have 15 different bivectors, however, this sub-algebra would also generate quadvectors and the hexvector for this space. So you'd end up with 32 distinct elements in that space.
However, a 4D Clifford Algebra has 6 bivectors and its one pseudo-scalar quadvector. This, together with its scalar, should generate Octonions.
And a 5D Clifford Algebra generates 10 bivectors and 5 quadvectors. Taking those plus scalar should be able to generate the Sedenions.
I did not actually test if those are equivalent, but that's the easiest way of how I could see those algebras.
I did test something quickly though: The subalgebra of even-dimensional subspaces of an n-D Clifford Algebra actually has dimension
Note that this trend also fits with the complex numbers. You have your elements {1, x, y, xy=I} of which the even sub-algebra is {1,I} and this happens to precisely model the complex numbers with the dimension
This would interestingly mean that, technically, octonions are made up not just of planar rotations (or double-reflections) but also of 4-space rotations (4x-reflections) and sedenions even have a full-space rotation (6 consecutive reflections in different directions) of sorts.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 27, 2014, 02:56:07 AM
In this (rather large, 268 pages) journal, there is a paper (starting page 101) showing how to properly do calculus with arbitrary non-commutative variables (as you might find them in geometric algebra, for instance)
http://ejtp.com/articles/ejtpv11i31.pdf#page=101
I haven't looked into all the other papers in this thing, so I can't say if anything else interesting is in there. Feel free to dig through it all :)
http://ejtp.com/articles/ejtpv11i31.pdf#page=101
I haven't looked into all the other papers in this thing, so I can't say if anything else interesting is in there. Feel free to dig through it all :)
Title: Programming through Rainy Days
Post by: hermann on September 27, 2014, 09:49:00 AM
Hallo Kram,
thanks for the information.
Some where it was stated that it was not possible to express octonions and sedonions but I can not remember the link.
The paper you posted looks very challanging. So I have my usual time constraints.
Also my holidays are know over and I am back in the company with challanging and time consuming jobs.
Normaly I prefere not to make software public before it is complete. But I discovered, that I can generate HTML-Documentation from GPS (Gnat Programing Studio).
So I have generated the documentation of my geometric algebra software to give a snapshot of the actual state of the work.
http://www.wackerart.de/mathematik/geometric_algebra.html#rainy_days (http://www.wackerart.de/mathematik/geometric_algebra.html#rainy_days)
It is still very incomplete, but the tables generated with this software can be seen on my geometric algebra page:
http://www.wackerart.de/mathematik/geometric_algebra.html (http://www.wackerart.de/mathematik/geometric_algebra.html) and in this thread.
I also made this incomplete work public to give you, jehovac and all in fractal forums a positive feedback for all the inspiration I have received.
Hermann
thanks for the information.
Some where it was stated that it was not possible to express octonions and sedonions but I can not remember the link.
The paper you posted looks very challanging. So I have my usual time constraints.
Also my holidays are know over and I am back in the company with challanging and time consuming jobs.
Normaly I prefere not to make software public before it is complete. But I discovered, that I can generate HTML-Documentation from GPS (Gnat Programing Studio).
So I have generated the documentation of my geometric algebra software to give a snapshot of the actual state of the work.
http://www.wackerart.de/mathematik/geometric_algebra.html#rainy_days (http://www.wackerart.de/mathematik/geometric_algebra.html#rainy_days)
It is still very incomplete, but the tables generated with this software can be seen on my geometric algebra page:
http://www.wackerart.de/mathematik/geometric_algebra.html (http://www.wackerart.de/mathematik/geometric_algebra.html) and in this thread.
I also made this incomplete work public to give you, jehovac and all in fractal forums a positive feedback for all the inspiration I have received.
Hermann
Title: Octonions
Post by: hermann on September 30, 2014, 09:03:18 PM
Octonions are not a Clifford algebra, since they are nonassociative.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 30, 2014, 11:41:34 PM
Oh, good point. Interesting.
That raises the question of how geometric algebraic "Octonions" (as proposed by me) compare to actual Octonions.
That raises the question of how geometric algebraic "Octonions" (as proposed by me) compare to actual Octonions.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on October 01, 2014, 05:38:31 AM
A good source on octonions in the internet is the page of John Baez:
http://math.ucr.edu/home/baez/octonions/ (http://math.ucr.edu/home/baez/octonions/)
May be I have some time next weekend, so that can I do some colouring on my geometric algebra tables. Also having different signatures may be helpful.
http://www.wackerart.de/mathematik/geometric_algebra.html (http://www.wackerart.de/mathematik/geometric_algebra.html)
Hermann
http://math.ucr.edu/home/baez/octonions/ (http://math.ucr.edu/home/baez/octonions/)
May be I have some time next weekend, so that can I do some colouring on my geometric algebra tables. Also having different signatures may be helpful.
http://www.wackerart.de/mathematik/geometric_algebra.html (http://www.wackerart.de/mathematik/geometric_algebra.html)
Hermann
Title: Cayley-Dickson Hypercomplex Numbers
Post by: hermann on October 05, 2014, 10:00:50 PM
I found this link:
Cayley-Dickson Hypercomplex Numbers Up to the Chingons (64D)
http://www.mapleprimes.com/posts/124913-Visualization-Of-The-CayleyDickson (http://www.mapleprimes.com/posts/124913-Visualization-Of-The-CayleyDickson)
Hermann
Cayley-Dickson Hypercomplex Numbers Up to the Chingons (64D)
http://www.mapleprimes.com/posts/124913-Visualization-Of-The-CayleyDickson (http://www.mapleprimes.com/posts/124913-Visualization-Of-The-CayleyDickson)
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on October 26, 2014, 12:10:43 AM
I've been playing around with general non-commutative multiplication lately, trying to push geometric algebra as far as possible, but I ran into an unexpected problem.
Say you have your n base-vectors.
Assume, multiplying them is non-commutative.
However, they are proportional to each other, differing only by a scalar factor
which commutes with everything:
From that first equation we also get
I.e. base-vectors always commute with themselves.
(Alternatively, the equation could also trivially be solved by setting
which would mean, that the given is a null-vector, i.e. its magnitude is 0.)
Furthermore, multiplying two base-vectors together gives rise to a bi-vector, which also is assumed to be proportional with yet another commutative scalar
:
etc.
That's probably all you can get out of just two indices. It becomes significantly more complicated once you get to three different indices.
Let's, for now, look at two different indices, one of which is duplicated. With that setup, there are three possibilities.
That last equation is a problem. There are three solutions to that equation:
This is weird. Either there is a significant error with how geometric algebra is setup (In normal geometric algebras,
, a possibility which I just demonstrated to lead to a paradox), or, which is a lot more likely, I made a major oversight somewhere in this. If somebody sees a problem, please point it out.
Say you have your n base-vectors
Assume, multiplying them is non-commutative.
However, they are proportional to each other, differing only by a scalar factor
From that first equation we also get
(Alternatively, the equation could also trivially be solved by setting
Furthermore, multiplying two base-vectors together gives rise to a bi-vector, which also is assumed to be proportional with yet another commutative scalar
etc.
That's probably all you can get out of just two indices. It becomes significantly more complicated once you get to three different indices.
Let's, for now, look at two different indices, one of which is duplicated. With that setup, there are three possibilities.
That last equation is a problem. There are three solutions to that equation:
- that'd be pretty useless
- in this case,
- that would mean, all base vectors commute. ⚡
This is weird. Either there is a significant error with how geometric algebra is setup (In normal geometric algebras,
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on October 26, 2014, 10:45:24 AM
I've been playing around with general non-commutative multiplication lately, trying to push geometric algebra as far as possible, but I ran into an unexpected problem.
On the first view I think you are mixing things you shouldn't mix.
Basicly in geometric algebra you have three products:
- The inner product
- The outer product
- And the geometric product
Now you define the inner product as anticommutative?
In geometric algebra you have to be very carefully if you multiply from left or right. (All so when producing the inverse).
Also take care with scalars.
Be carefull in geometric algebra you have different products! Don't mix them in a way you are trained for products in school.
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on October 26, 2014, 11:15:41 AM
I was actually using the geometric product.
The geometric product of a base vector with itself is commutative (it's what the inner product gives you, and in fact it must be in the way I framed it above:
is easily shown to always be 1, i.e. changing order does not result in sign change, i.e. it's commutative.), but with another base vector it's anti-commutative (xy = -xy etc.), which is the wedge-part of the product.
And scalars are completely commutative.
That's what I was assuming here. Or at least I was trying to.
I still must have made some error of that kind somewhere in there. But I can't really spot precisely where.
Basically, all I did was generalizing two things in Geometric Algebra to support arbitrary scalars: Switching (geometric) multiplication orders of base-vectors
results in a swap-factor 
and (geometrically) multiplying two vectors 
results in another scalar factor 
which is essentially the signature, except I have used it on all base-vectors, not just ones of equal index. (Normally, the signature would appear like this: 
; where 
simply is the signature for that particular vector. I have added the possibility for giving mixed vectors such a signature factor as well.)
Nowhere in this am I even using the scalar product or the wedge product, except for where they happen to align with the geometric product.
All I was trying to do is to see for which instances of and this is consistent; if, perhaps, this only even consistently works for )
and )
I.e. whether this only works for the conditions we already know about (swapping unequal vectors anticommutes, swapping equal ones commutes, only base 1-vectors get an additional factor through signature) or whether there are more general situations.
Now it would be no problem if what I showed was, that only what we already do is allowed, but if I made no mistake (and I probably did), it means that not even that case is allowed.
Edit: Tex doesn't work well with the notation I used up there in c. What I was trying to convey is that c could have an arbitrary list of indices in general, for products of arbitrary n-vectors with arbitrary k-vectors. I wasn't quite sure how else to depict that, but this method using products clearly doesn't work all too well.
The geometric product of a base vector with itself is commutative (it's what the inner product gives you, and in fact it must be in the way I framed it above:
And scalars are completely commutative.
That's what I was assuming here. Or at least I was trying to.
I still must have made some error of that kind somewhere in there. But I can't really spot precisely where.
Basically, all I did was generalizing two things in Geometric Algebra to support arbitrary scalars: Switching (geometric) multiplication orders of base-vectors
Nowhere in this am I even using the scalar product or the wedge product, except for where they happen to align with the geometric product.
All I was trying to do is to see for which instances of
I.e. whether this only works for the conditions we already know about (swapping unequal vectors anticommutes, swapping equal ones commutes, only base 1-vectors get an additional factor through signature) or whether there are more general situations.
Now it would be no problem if what I showed was, that only what we already do is allowed, but if I made no mistake (and I probably did), it means that not even that case is allowed.
Edit: Tex doesn't work well with the notation I used up there in c. What I was trying to convey is that c could have an arbitrary list of indices in general, for products of arbitrary n-vectors with arbitrary k-vectors. I wasn't quite sure how else to depict that, but this method using products clearly doesn't work all too well.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on October 26, 2014, 07:43:13 PM
Partial products are defined from "the product"...a la from Lie algebra. (ab-ba)/2, (ab+ba)/2...toss in ops like conjugates to define more.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on October 26, 2014, 08:59:33 PM
I am aware.
Here:
Geometrically multiplying a base vector with itself is the same as the scalar product of that base vector.
 c_{ij}}{2} e_{ij})
If
(as it is for "normal" GA), this would become 0.
 c_{ij}}{2} e_{ij})
If (as it is for "normal" GA), this would become  c_{ij} }{2} e_{ij} = c_{ij} e_{ij} )
.
Geometrically multiplying two different base-vectors with each other is the same as the wedge product of those vectors (or at least it would be if
, as it normally is)
Throughout the whole thing I was using the geometric product.
The only two differences are that I am allowing for some arbitrary additional scalar factor appearing even after multiplying vectors geometrically, which are not the same (), and that I'm allowing non-commutative multiplication to result in yet another arbitrary factor (
).
Here:
Geometrically multiplying a base vector with itself is the same as the scalar product of that base vector.
If
If
Geometrically multiplying two different base-vectors with each other is the same as the wedge product of those vectors (or at least it would be if
Throughout the whole thing I was using the geometric product.
The only two differences are that I am allowing for some arbitrary additional scalar factor appearing even after multiplying vectors geometrically, which are not the same (
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on October 27, 2014, 07:35:29 AM
I wasn't clear...that was a reply to hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on October 28, 2014, 09:14:29 PM
(http://www.wackerart.de/mathematik/geometric_algebra/bivector.gif)
It is an area with an assigned orientation.
If you first move the distance
I think you have changed the basic properties of the outer product a little bit too much. Or do you have a meaningfull geometric interpretation in the plane?
In many physical theories path independence is very imortant. I have the impression, that this property gets lost when you define an outer product in this way.
I have not inspected the algebraic structres in detail may be this can be interesting.
Title: Geometric Product
Post by: hermann on October 28, 2014, 09:50:20 PM
The geometric product is defined as follows:
It contains a commutative part, the dot product or inner product and the anticommutative part called wedge product or outerproduct.
If you define the inner product also anticommutaive or with a factor you get something like this:
Hermann
It contains a commutative part, the dot product or inner product and the anticommutative part called wedge product or outerproduct.
If you define the inner product also anticommutaive or with a factor you get something like this:
Hermann
Title: Lie-Algebra
Post by: hermann on October 28, 2014, 09:57:37 PM
I think a Lie-Algebra is also close to this issue:
![[a,b] = ab - ba](/cgi-bin/mimetex.cgi?[a,b] = ab - ba)
But I am not an expert in Lie-Algebra.
But I am not an expert in Lie-Algebra.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on October 28, 2014, 10:01:04 PM
What about:
using the geometric product?
using the geometric product?
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on October 28, 2014, 10:05:43 PM
Ortogonalty of the base vectors requires:
if i and j are not equal!
if i and j are not equal!
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on October 29, 2014, 12:23:56 AM
Thanks for those contributions :)
At this point I am just experimenting.
I'm not even concerned with the outer product right now. As said, I only experimented with the geometric product.
However, you are right about the outer product being changed quite significantly.
As I already showed in my previous post, in using a
, you actually don't even get a pure antisymmetric part. You get both a scalar (which is symmetric) AND a(n antisymmetric) bi-vector component if you use the standard definition of a wedge product.
I did not define the inner product as anti-commutative. I already showed how, i.e. the factor you get from exchanging a base-vector with itself, must always be 1. Hence, it is symmetric.
, that is correct. It's a nice alternate method to denote it.
Your orthogonality remark is topical: That's actually why I am investigating this direction.
You do not actually need an orthonormal basis of vectors. You can just have n completely arbitrary base-vectors for an n-dimensional space. They can lie at completely odd angles to each other. The only limitation is that they must not be linearly dependent on each other.
That
would just be a generalization. Nothing unheard of. In fact, in certain cases, like, for instance, in triclinic crystals (http://en.wikipedia.org/wiki/Triclinic_crystal_system), it's actually more natural to describe your coordinates in such a non-orthognonal and perhaps even non-normal base.
However, I have found the problem with my problem now. I did nothing wrong in my calculations, but I interpreted it wrong:
The offending equation was the following:
So that means:
I previously simply assumed, that that means, could only be 1 to fullfill this equation. However, all that is required, in accordance with a different equation I already got before, is this:
and
This is required for sake of consistency.
Thus, the whole system reduces again to precisely what we already had before, as far as commutativity goes.
All that is left is our, which now are restricted a slight bit further too.
In particular:
From there we could get:
This is definitely consistent if we just set
. I'm not absolutely certain right now, that this is the only consistent concept, but it might be.
If it turns out that this is the unique solution, we'd essentially have some symmetric matrix
of which each 
and 
stands for the additional factor of a geometric product of the 
and 
base vectors. - This would essentially be the metric tensor of our given system. The main diagonal would give the usual signature of the system, while all the other values 
correspond to rotated components. They only occur, if the system is not orthogonal.
As said, I'm not yet fully convinced, that this is the only solution here, though it's not unlikely.
But if it is, we'll still have to look into what happens for other vector multiplications. For instance, what happens if we multiply a vector and a bi-vector or, essentially equivalently, three vectors? Things can become really complicated really quickly, but most certainly, a pattern for multiplying
with 
vectors should appear sooner rather than later.
At this point I am just experimenting.
I'm not even concerned with the outer product right now. As said, I only experimented with the geometric product.
However, you are right about the outer product being changed quite significantly.
As I already showed in my previous post, in using a
I did not define the inner product as anti-commutative. I already showed how
Your orthogonality remark is topical: That's actually why I am investigating this direction.
You do not actually need an orthonormal basis of vectors. You can just have n completely arbitrary base-vectors for an n-dimensional space. They can lie at completely odd angles to each other. The only limitation is that they must not be linearly dependent on each other.
That
However, I have found the problem with my problem now. I did nothing wrong in my calculations, but I interpreted it wrong:
The offending equation was the following:
So that means:
I previously simply assumed, that that means,
Thus, the whole system reduces again to precisely what we already had before, as far as commutativity goes.
All that is left is our
In particular:
This is definitely consistent if we just set
If it turns out that this is the unique solution, we'd essentially have some symmetric matrix
As said, I'm not yet fully convinced, that this is the only solution here, though it's not unlikely.
But if it is, we'll still have to look into what happens for other vector multiplications. For instance, what happens if we multiply a vector and a bi-vector or, essentially equivalently, three vectors? Things can become really complicated really quickly, but most certainly, a pattern for multiplying
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on October 29, 2014, 04:30:14 AM
if the do not form an orthogonal base I think you have to play the game with covariant and contravariant vectors and you have to define what your metric is.
(http://www.wackerart.de/mathematik/affine-koordinaten/e-1-kontra.gif)
)
being the spat product!
I have not tried to write this down with geometric algebra or do we have a source where such thinks are discibed?
(http://www.wackerart.de/mathematik/affine-koordinaten/e-1-kontra.gif)
I have not tried to write this down with geometric algebra or do we have a source where such thinks are discibed?
Title: Re: Geometric Product
Post by: Roquen on October 29, 2014, 02:52:26 PM
The geometric product is defined as follows:
<Quoted Image Removed>
To be anal, no it's not defined that way...the product rules for the basis elements define the full product. This is an identity to show these two specific partial products are related to the product.<Quoted Image Removed>
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on October 29, 2014, 10:05:20 PM
Not really.
You could define
just as well as you could define 
It's almost the same thing as defining:
versus 
In both cases I'd argue that the second version is the more fundamental one, however, there is not actually anything wrong with doing it like in the first version. You just have to pick.
One of the two things is defined, and the other is derived. Which one matters very little.
Really, this can be expressed as:
and
respectively. In either case it takes simple algebraic manipulations to show that these are correct.
You could define
It's almost the same thing as defining:
In both cases I'd argue that the second version is the more fundamental one, however, there is not actually anything wrong with doing it like in the first version. You just have to pick.
One of the two things is defined, and the other is derived. Which one matters very little.
Really, this can be expressed as:
and
respectively. In either case it takes simple algebraic manipulations to show that these are correct.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on October 29, 2014, 10:24:35 PM
Thanks Kram,
I am a little tired and exhausted this evening so I can work it out in detail.
But I think your construction has something to do with the metric tensor which plays an importend role in general relativety.
http://de.wikipedia.org/wiki/Metrischer_Tensor (http://de.wikipedia.org/wiki/Metrischer_Tensor)
http://de.wikipedia.org/wiki/Krummlinige_Koordinaten (http://de.wikipedia.org/wiki/Krummlinige_Koordinaten)
Hermann
P.S may be I can work out the details the next days.
I am a little tired and exhausted this evening so I can work it out in detail.
But I think your construction has something to do with the metric tensor which plays an importend role in general relativety.
http://de.wikipedia.org/wiki/Metrischer_Tensor (http://de.wikipedia.org/wiki/Metrischer_Tensor)
http://de.wikipedia.org/wiki/Krummlinige_Koordinaten (http://de.wikipedia.org/wiki/Krummlinige_Koordinaten)
Hermann
P.S may be I can work out the details the next days.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on October 29, 2014, 11:28:48 PM
Indeed it does, and I already said so
Btw, why do you link to the German Wikis? - Obviously, you and I know German. But I think the largest part of the community, by language, actually is natively English speaking and doesn't understand German either well or at all.
http://en.wikipedia.org/wiki/Metric_tensor
http://en.wikipedia.org/wiki/Curvilinear_coordinates
(Also, very often, English wikis are better than their German counterparts. - Although the German Wikipedia is amongst the best. Especially in Chemistry it actually frequently beats the English one in my experience. Still, usually, English is better, and it's more universally understood here :))
This would essentially be the metric tensor of our given system.
:DBtw, why do you link to the German Wikis? - Obviously, you and I know German. But I think the largest part of the community, by language, actually is natively English speaking and doesn't understand German either well or at all.
http://en.wikipedia.org/wiki/Metric_tensor
http://en.wikipedia.org/wiki/Curvilinear_coordinates
(Also, very often, English wikis are better than their German counterparts. - Although the German Wikipedia is amongst the best. Especially in Chemistry it actually frequently beats the English one in my experience. Still, usually, English is better, and it's more universally understood here :))
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on October 31, 2014, 11:39:51 PM
http://wavewatching.net/2014/10/27/the-unintentional-obsfuscation-of-physics/
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on November 01, 2014, 01:20:25 AM
Here's another wild idea. I didn't actually test it much, yet. Could be complete nonsense.
However, as we have seen, only had the following condition on it: 
.
I assumed sigma to be a scalar. But what happens if you let it be its own vector?
For instance, starting with a 2D Geometric Algebra with positive signature:
^2= (a^2 + b^2 + c^2 - d^2) + 2 a b x + 2 a c y + 2 a d i=1)
This gives the solution:
 \vee \left(b=c=d=0\wedge a^2=1\right))
Anything that sticks to those parameters,
- is its own inverse
- should be a plausible definition of.
This kind of extension can become very complicated very quickly though. It remains to be seen whether that would even be a valuable addition.
However, as we have seen,
I assumed sigma to be a scalar. But what happens if you let it be its own vector?
For instance, starting with a 2D Geometric Algebra with positive signature:
This gives the solution:
Anything that sticks to those parameters,
- is its own inverse
- should be a plausible definition of
This kind of extension can become very complicated very quickly though. It remains to be seen whether that would even be a valuable addition.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on November 01, 2014, 04:57:54 PM
Here is an awesome paper: Vector Analysis of Spinors (pdf) (http://www.garretstar.com/vecanal27-10-2014.pdf)
It's pretty simple to follow as far as I've seen.
I haven't even reached the meat of the paper yet (I'm just about to get there), but already what happens before that is pretty nice.
It's not very well checked though. I found a few errors: Two times, early on, there is a
which should be a 
, later down the line, a 
in an exponent is supposed to be an and one of the worst errors I found thus far is falsely calling Heisenberg Heisenburg.
There also is this flash video talk:
http://www.worldsci.org/php/DimDimFlashViewer.php?id=336
I wish the quality would be a bit higher. He sadly isn't a great talker: He's very clearly nervous. Though it's still a good talk.
The slides for that talk can be found here http://garretstar.com/nfmtalk2010/AMS-MAA2013.pdf
It's pretty simple to follow as far as I've seen.
I haven't even reached the meat of the paper yet (I'm just about to get there), but already what happens before that is pretty nice.
It's not very well checked though. I found a few errors: Two times, early on, there is a
There also is this flash video talk:
http://www.worldsci.org/php/DimDimFlashViewer.php?id=336
I wish the quality would be a bit higher. He sadly isn't a great talker: He's very clearly nervous. Though it's still a good talk.
The slides for that talk can be found here http://garretstar.com/nfmtalk2010/AMS-MAA2013.pdf
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on November 02, 2014, 01:29:21 AM
Apparently, this paper of his is more recent:
Geometry of Spin-
-particles (http://www.garretstar.com/geospin08-6-14.pdf)
He apparently was a direct grad student of Hestenes.
Geometry of Spin-
He apparently was a direct grad student of Hestenes.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on November 02, 2014, 08:17:33 PM
Interesting links!
I hope I have more time next weekend to look on it in detail.
Hermann
I hope I have more time next weekend to look on it in detail.
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on November 02, 2014, 08:58:15 PM
It's actually pretty timely: My recent experiments partially aimed at answering exactly the questions he answered in those papers (if not in full generality). The gist of his work appears to be to unify matrices and Geometric Algebra to the point where you can do everything with GA.
He's focusing heavily on idempotents (values
which have the property, that 
. For the reals, this only works with 
and ) and nilpotents (
) and, very importantly, he establishes, that such idempotents pretty much are the Geometric Algebra version of Eigenvalues.
This is something I have pondered about for a while, actually. Eigenvalues and Eigenvectors are, like, THE most important thing behind matrices. If you have them for a given matrix, you can apply /any/ analytic function to those matrices.
Now what is left for me to figure out is: what about non-square matrices, and how to fully translate arbitrary tensor algebra into this format?
It certainly must be possible, by way of the underlying isomorphisms.
Sadly, he only touches on square matrices, focusing on 2x2- and 3x3-ones, not really saying much about nxn- and saying nothing about nxk matrices.
However, a part of what he did hinted at possible extensions for such arbitrary cases.
Obviously, an nxk matrix represents an either under- or over-determined problem. I was wondering how well GA could deal with such cases. I haven't found a whole lot on that yet.
And the other thing, tensor-algebra, is because we learn it that way at university. Coordinate transformations from one frame to another appear to be half of the bread and butter of a physicist. And even if Tensor-algebra is highly cumbersome at times, especially once you get into Co- vs. Contravariant stuff (damn that stuff is easy to mix up. Luckily the distinction isn't always important), it's still really powerful.
I've seen demonstrations of specific cases, but I haven't yet seen a generic 1:1 translation of something expressed with arbitrary tensors to something expressed in some fitting Geometric Algebra, even if that is, alledgedly, completely possible.
Once we can have arbitrary-dimensioned tensors fully translated in GA and back, we can truly compare the two approaches. (Thus far, what I've seen is mostly "very promising", to the point where I'd love to just do everything this way, but no complete, rigorous, exhaustive 1:1-translation)
He's focusing heavily on idempotents (values
This is something I have pondered about for a while, actually. Eigenvalues and Eigenvectors are, like, THE most important thing behind matrices. If you have them for a given matrix, you can apply /any/ analytic function to those matrices.
Now what is left for me to figure out is: what about non-square matrices, and how to fully translate arbitrary tensor algebra into this format?
It certainly must be possible, by way of the underlying isomorphisms.
Sadly, he only touches on square matrices, focusing on 2x2- and 3x3-ones, not really saying much about nxn- and saying nothing about nxk matrices.
However, a part of what he did hinted at possible extensions for such arbitrary cases.
Obviously, an nxk matrix represents an either under- or over-determined problem. I was wondering how well GA could deal with such cases. I haven't found a whole lot on that yet.
And the other thing, tensor-algebra, is because we learn it that way at university. Coordinate transformations from one frame to another appear to be half of the bread and butter of a physicist. And even if Tensor-algebra is highly cumbersome at times, especially once you get into Co- vs. Contravariant stuff (damn that stuff is easy to mix up. Luckily the distinction isn't always important), it's still really powerful.
I've seen demonstrations of specific cases, but I haven't yet seen a generic 1:1 translation of something expressed with arbitrary tensors to something expressed in some fitting Geometric Algebra, even if that is, alledgedly, completely possible.
Once we can have arbitrary-dimensioned tensors fully translated in GA and back, we can truly compare the two approaches. (Thus far, what I've seen is mostly "very promising", to the point where I'd love to just do everything this way, but no complete, rigorous, exhaustive 1:1-translation)
Title: Bivectors
Post by: hermann on November 04, 2014, 10:57:53 PM
I found this fine article on Bivectors in Wikipedia:
http://en.wikipedia.org/wiki/Bivector (http://en.wikipedia.org/wiki/Bivector)
Hermann
http://en.wikipedia.org/wiki/Bivector (http://en.wikipedia.org/wiki/Bivector)
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on November 06, 2014, 01:16:35 AM
Thanks Hermann :)
There's also http://en.wikipedia.org/wiki/Multivector
Meanwhile, just for fun, idempotents in spherical coordinates:
with
then = u_\pm^2 \Rightarrow \\<br />u_0 = u_0^2+u_r^2 e_r^2+u_\theta^2 e_\theta^2+u_\phi^2 e_\phi^2 = u_0^2+u_r^2+r^2 u_\theta+\left({r \sin \theta}\right)^2 u_\phi^2 \\<br />u_r = 2 u_0 u_r \\<br />u_\theta = 2 u_0 u_\theta \\<br />u_\phi = 2 u_0 u_\phi \Rightarrow \\<br />u_0=\frac{1}{2} \Rightarrow \\<br />\frac{1}{2} = \frac{1}{4} + u_r^2+r^2 u_\theta+\left({r \sin \theta}\right)^2 u_\phi^2 \Rightarrow \\<br />u_r^2+r^2 u_\theta+\left({r \sin \theta}\right)^2 u_\phi^2 = \frac{1}{4} \\<br />u_\theta^2+\sin^2\theta \: u_\phi^2 =\frac{\frac{1}{4}-u_r^2}{r^2}<br />)
You could substitute that back into the orignal, but I think it's nicer to look at with that condition. As long as
and 
, we are looking at an idempotent, squaring to itself.
There's also http://en.wikipedia.org/wiki/Multivector
Meanwhile, just for fun, idempotents in spherical coordinates:
with
then
You could substitute that back into the orignal, but I think it's nicer to look at with that condition. As long as
Title: Re: Geometric Algebra, Geometric Calculus
Post by: Roquen on November 06, 2014, 07:09:22 AM
Quote
In both cases I'd argue that the second version is the more fundamental one, however, there is not actually anything wrong with doing it like in the first version. You just have to pick.
One of the two things is defined, and the other is derived. Which one matters very little.
Like I said: I was being anal. But if you define in terms of the partial products then number of axioms explodes. The simplest definitions wins.One of the two things is defined, and the other is derived. Which one matters very little.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on November 08, 2014, 04:13:25 PM
Ok, digging through Garret Sobczyk's older papers (http://scholar.google.com/citations?hl=en&user=yoQ7uDcAAAAJ&view_op=list_works&sortby=pubdate), I figured this out a bit more, though I'm still not entirely sure how it works.
His whole point is that, using his propsed basis of idempotents and nilpotents - the "generalized spectral basis" -, the product table simplifies itself a lot.
For instance, if I'm not mistaken (calculated all those myself) here are all of his proposed nilpotents and idempotents in the Clifford-Algebra with signature
:
 \\<br />s2_\phi = s2_\phi^2 = \frac{1}{2} \left( 1 - x \cos{\phi} - y \sin{\phi} \right) \\<br />s1_\phi s2_\phi = 0 \\<br />s1_\phi + s2_\phi = 1 \\<br />q1_\phi = s1_\phi q1_\phi = r \left(x \sin{\phi} + y \cos{\phi} + i \right) \\<br />q2_\phi = s2_\phi q2_\phi = r \left(x \sin{\phi} - y \cos{\phi} + i \right) \\<br />q1_\phi^2 = q2_\phi^2 = 0 \\<br />q1_\phi q2_\phi = 2 r^2 \left(\sin{\phi}^2-\left(x+i \sin{\phi} \right)\cos{\phi}\right))
All but the very last one of those are rather pretty. I'm not quite sure what to make of that last one. Here's a multiplication table:
I didn't yet try out all of those. All of those which have two terms in them should, according to Sobczyk, either cancel to 0 or become one of the original terms. That's why I'm a little confused by the rather strange-looking form of
, though it might be that I'll actually have to give up some more generality, fixing either the 
or the 
or perhaps both to get to that especially nice form. Still reading to figure that out.
Speaking of generality: for all four terms,
, I started with a general object containing all base-
-vectors, so if something is missing, it's because some condition on those objects requires that it is 0.
His whole point is that, using his propsed basis of idempotents and nilpotents - the "generalized spectral basis" -, the product table simplifies itself a lot.
For instance, if I'm not mistaken (calculated all those myself) here are all of his proposed nilpotents and idempotents in the Clifford-Algebra with signature
All but the very last one of those are rather pretty. I'm not quite sure what to make of that last one. Here's a multiplication table:
I didn't yet try out all of those. All of those which have two terms in them should, according to Sobczyk, either cancel to 0 or become one of the original terms. That's why I'm a little confused by the rather strange-looking form of
Speaking of generality: for all four terms,
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on November 10, 2014, 08:58:08 PM
I appear to have miscalculated . It actually becomes )
and the product )
and 
Meanwhile,\left(q_2 q_1\right) = 4 r^4 \sin{2\phi} \left(\sin\phi - y\right))
?
I must be doing something wrong because, if that is true, associativity would no longer hold, which would be kinda awkward. Quite clearly, from
, \left(q_2 q_1\right) = q_1\left(q_2^2\right)q_1 = q_1 0 q_1 = 0 \neq 4 r^4 \sin{2\phi} \left(\sin\phi - y\right))
...
Does anybody see where my error lies?
Meanwhile,
I must be doing something wrong because, if that is true, associativity would no longer hold, which would be kinda awkward. Quite clearly, from
Does anybody see where my error lies?
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on November 12, 2014, 12:51:37 AM
Ok, things are more complicated than I thought they would be.
While
, it is not true, that 
. Thus, we are in the weird situation, where 
acts as some kind of non-commutative unity. That's quite like if 
.
However, I can't actually find references in Garret Sobczyk's works, where he actually uses the nilpotents
in his GA basis. They don't appear to always occur or, more accurately, they often appear to just be . So perhaps this actually is an instance of that.
In particular, if I require
, that requires me to set 
, so 
.
In that case I actually only have to deal with the idempotents
and the entire product-table becomes:
Interestingly, while
and 
, it still is true that 
.
This is an important property. It, for instance, means, that^k = (M_1 s_1 + M_2 s_2)^k = M_1^k s_1 + M_2^k s_2)
(for some arbitrary Multivector 
)
and through applying this idea to a Taylor series, that also means that = f\left(M_1 s_1+M_2 s_2\right) = f\left(M_1\right) s_1 + f\left(M_2\right) s_2)
. This is what makes this basis so powerful. - Assuming you can easily find the value of a given function at 
, you can easily calculate any function at some general M.
That's all assuming that I got that right and thes are indeed 0 for this problem. If they actually are not, that means a general function would somehow use them.
A bit more explanation:
Since idempotents have the following propery:
, they act as projectors. i.e. if you multiply something with them muliple times, that thing will no longer change. 
- that's what I used above when I said 
. The very first product changes things around, but every further product doesn't change a thing.
Similarly, since implies 
and something of the form ^k)
will produce some form that has every combination )
, we get, for all those terms, except for when 
or 
, simply the replacement term 
. This term, however, by the properties of our special idempotents, is . Thus only the two terms with the highest powers remain. Those two leftover terms reduce to and 
respectively.
If you put all this into a taylor series, you'll get the above result, that any analytic function can be expressed by.
Lastly, there is one more weird thing about these idempotents:
would be the "reflection" of in . I put that in "quotation marks", because it only really would be a reflection if I used pure vectors, and I obviously am using vectors with a scalar part here.
This reflection completely knocks out the bivector part. Furthermore, no matter what side you multiply onto that product, the result will no longer change:
 + y \cos\theta \left(m_0+m_2\cos\theta+m_1\sin\theta\right)\right))
So for this value, "reflection" and projection kind of coincide. I wonder about the exact geometric interpretation of this.
While
However, I can't actually find references in Garret Sobczyk's works, where he actually uses the nilpotents
In particular, if I require
In that case I actually only have to deal with the idempotents
Interestingly, while
This is an important property. It, for instance, means, that
and through applying this idea to a Taylor series, that also means that
That's all assuming that I got that right and the
A bit more explanation:
Since idempotents have the following propery:
Similarly, since
If you put all this into a taylor series, you'll get the above result, that any analytic function can be expressed by
Lastly, there is one more weird thing about these idempotents:
This reflection completely knocks out the bivector part. Furthermore, no matter what side you multiply
So for this value, "reflection" and projection kind of coincide. I wonder about the exact geometric interpretation of this.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: jehovajah on November 12, 2014, 09:30:26 AM
Thanks for the links. Kram1032. I have added them to my collection of links on this topic.
I find Norman Wildberger a good source for most of these fundamentals of the topic. I like to look at the different presentations though because it gives me some alternative views on the meaning or possible interpretation of Hermann Grassmanns Labels or Handles or Notions(Begriffe).
Keep up the good Work and wishing you success in your studies and research.
Hermann I also thank you for your interest and contribution. The issue of visualisation and dimensions is a tricky one. Currently I go with crystal spaces and facets as realisations of multidimensional extensive magnitudes. I am a big Scirnce fiction fan, but draw the line when it comes to hidden other worldly dimensions.
Hey, anything is possible, right? certainly the fractal generator allows us to tour a fractal in all these possible dimensioned spaces, with boundaries materialising and dematerialising as we go, but pragmatically I do not have time to figure out other solutions. Confusing as it is I just make do with interleaving crystal spaces.
V9 and V18 are an attempt to understand the geometrical model underpinning Vortex Based Mathrmatics as exposited by Randy Powell. I was and am curious to see what a fractal generator makes of it. I was encouraged somewhat by my design of Newtonian Triples, where the dimensions were directly manageable by a quaternion framed fractal generator.
However, since I did not understand or rather "everyway stand" what I was doing I thought it best to get to grips with Hermann Grassmanns methods and prior to that Hamiltons. What a treat that has turned out to be!
Yes I will eventually finish the V9 " *" table and the V18, but I want to know what the Fractsl images might relate to in magneto dynamics , say, not only just for the hopefully beautiful forms!
I find Norman Wildberger a good source for most of these fundamentals of the topic. I like to look at the different presentations though because it gives me some alternative views on the meaning or possible interpretation of Hermann Grassmanns Labels or Handles or Notions(Begriffe).
Keep up the good Work and wishing you success in your studies and research.
Hermann I also thank you for your interest and contribution. The issue of visualisation and dimensions is a tricky one. Currently I go with crystal spaces and facets as realisations of multidimensional extensive magnitudes. I am a big Scirnce fiction fan, but draw the line when it comes to hidden other worldly dimensions.
Hey, anything is possible, right? certainly the fractal generator allows us to tour a fractal in all these possible dimensioned spaces, with boundaries materialising and dematerialising as we go, but pragmatically I do not have time to figure out other solutions. Confusing as it is I just make do with interleaving crystal spaces.
V9 and V18 are an attempt to understand the geometrical model underpinning Vortex Based Mathrmatics as exposited by Randy Powell. I was and am curious to see what a fractal generator makes of it. I was encouraged somewhat by my design of Newtonian Triples, where the dimensions were directly manageable by a quaternion framed fractal generator.
However, since I did not understand or rather "everyway stand" what I was doing I thought it best to get to grips with Hermann Grassmanns methods and prior to that Hamiltons. What a treat that has turned out to be!
Yes I will eventually finish the V9 " *" table and the V18, but I want to know what the Fractsl images might relate to in magneto dynamics , say, not only just for the hopefully beautiful forms!
Title: Re: Geometric Algebra, Geometric Calculus
Post by: jehovajah on November 12, 2014, 10:27:17 AM
Actually it just came back to mind. I remember setting Laz Plaths, aka qqazxxsw on YouTube , trochoid applications as my favourite visualisation tools! You can set up an n dimensional trochoid very easily and intuitively on his apps, because each rotating circle represents a dimension.
Check out his channel and videos. Count how many circles he uses!
I honestly think this man has solved the problem of visualising n dimensions in every practicable way. But he won't talk to me!
Oh well, that's life I suppose :D
Check out his channel and videos. Count how many circles he uses!
I honestly think this man has solved the problem of visualising n dimensions in every practicable way. But he won't talk to me!
Oh well, that's life I suppose :D
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on November 23, 2014, 07:23:21 PM
Spacetime Algebra for Electromagnetism (http://arxiv.org/pdf/1411.5002.pdf)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on December 04, 2014, 02:14:21 PM
I'm not sure if I already linked this paper here (slowly but surely the thread becomes kinda long to check every single post)
Geometric Algebra - Reformulation of General Relativity (http://arxiv.org/pdf/gr-qc/0311007v1.pdf) - this one will obviously not be suitable for beginners: Even if GA is able to simplify geometric formulations, or rather, it adds nice geometric insight in abstract algebraic formulae, GR is so abstract, it'll take getting used to in any kind of formulation.
That being said, I personally, having at least a vague idea of tensor calculus, can follow at the very least the first section (Everything up to and including the part where they solve the Schwarzschild-Metric) with some success. The most difficult bit is getting used to the billions of symbols. But once you accept their meaning, it's fine.
Geometric Algebra - Reformulation of General Relativity (http://arxiv.org/pdf/gr-qc/0311007v1.pdf) - this one will obviously not be suitable for beginners: Even if GA is able to simplify geometric formulations, or rather, it adds nice geometric insight in abstract algebraic formulae, GR is so abstract, it'll take getting used to in any kind of formulation.
That being said, I personally, having at least a vague idea of tensor calculus, can follow at the very least the first section (Everything up to and including the part where they solve the Schwarzschild-Metric) with some success. The most difficult bit is getting used to the billions of symbols. But once you accept their meaning, it's fine.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on January 17, 2015, 12:58:01 PM
Here are the slides and exercises for a complete course about Computing with Geometric Algebra http://www.gaalop.de/ga-computing-lecture/
It's by the guy behind the Gaalop Geometric Algebra package (hence the website), who also works at TU Darmstadt.
EDIT: apparently the slides are a mixture of English and German.
It's by the guy behind the Gaalop Geometric Algebra package (hence the website), who also works at TU Darmstadt.
EDIT: apparently the slides are a mixture of English and German.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on January 27, 2015, 10:27:14 AM
A new approach to euclidean plane geometry based on projective geometric algebra (http://arxiv-web3.library.cornell.edu/abs/1501.06511v1)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: jehovajah on January 27, 2015, 10:30:52 PM
Thanks for the link Kram1032. :dink:
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on February 17, 2015, 03:02:06 PM
I've read a lot about a general Multivector not having an inverse and it not being possible to define one.
However I played around and found a way for at least the normal positive signature
-case. If you just want the formula, jump to the end. What follows is an explanation of how I arrived at the result (plus background you probably already know if you follow this thread. I wanted to make this post fairly self-contained).
To define an inverse for complex numbers, what you have to do is this:
Essentially, to find a division, you have to figure out a way to make the denominator a real number just by multiplying numbers which are equal to 1.
In case of the complex number above, what you have to do is multiply by
, requiring a single multiplication to get to the desired result.
Now, it indeed doesn't appear to be possible to do a single multiplication to get to a real number (i.e. a pure scalar) but multiple multiplications do the trick.
First, let me define a few short-hand notations:
You can write a Multivector)
in one of the following equivalent notations: =\left(a,\left(x_1,x_2,x_3\right),\left(B_1,B_2,B_3\right) i,c i\right) = \\ a+x_1 e_1 + x_2 e_2+x_3 e_3 + B_1 e_2 e_3 + B_2 e_3 e_1 + B_3 e_1 e_2 +c e_1 e_2 e_3 = \left(\left(a+c i\right)+\left(x_1+B_1 i\right)e_1+\left(x_2+B_2 i\right)e_2+\left(x_3+B_3 i\right)e_3\right) = \\ \left(a+i c,\mathbf{x}+i\mathbf{B}\right) = \left(a+i c,\left(x_1+i B_1,x_2+i B_2,x_3+i B_3\right)\right))
For convenience's sake, I'll be using the second to last notation.
I'll furthermore introduce the shorthand
and 
So then my general Multivector becomes)
Although here is technically a shorthand for 
, in the following it can be treated exactly like the familiar imaginary unit.
is the normal (real) dot product. (Do not conjugate complex numbers)
is the normal cross product.
The wedge product
The Geometric Product of two Multivectors
is defined as follows:
\right))
Then there is the Flip operation which simply switches the sign of the vector part:
}^F =\left(\alpha,-\mathbf{\nu}\right) = \left(a+i c,-\mathbf{x}-i\mathbf{B}\right) )
and the Conjugate, which switches the sign of the imagnary part:
}^* =\left(\alpha^*,\mathbf{\nu}^*\right) = \left(a-i c,\mathbf{x}-i\mathbf{B}\right))
Ok, with all this we can finally define inversion:
^*}{\left(M M^F\right)^*} = \frac{M^F \left(M M^F\right)^*}{M M^F \left(M M^F\right)^*)
For this last fraction, the denominator is indeed real-valued. The given inverse is both a left- and a right-inverse, so no special care has to be taken for the order.
I.e.
Finally, here is the result expanded in full component form (no longer with real denominator, but as said, you can treat like you would treat any complex number):
\right) \frac{1}{{\left(a+i c\right)}^2-\left({\left(x_1+i B_1\right)}^2+{\left(x_2+i B_2\right)}^2+{\left(x_3+i B_3\right)}^2\right)})
Obviously, there are some problems with multivectors for which}^2-\left({\left(x_1+i B_1\right)}^2+{\left(x_2+i B_2\right)}^2+{\left(x_3+i B_3\right)}^2\right)}=0)
. These are the null multivectors for which division is indeed undefined.
However I played around and found a way for at least the normal positive signature
To define an inverse for complex numbers, what you have to do is this:
Essentially, to find a division, you have to figure out a way to make the denominator a real number just by multiplying numbers which are equal to 1.
In case of the complex number above, what you have to do is multiply by
Now, it indeed doesn't appear to be possible to do a single multiplication to get to a real number (i.e. a pure scalar) but multiple multiplications do the trick.
First, let me define a few short-hand notations:
You can write a Multivector
For convenience's sake, I'll be using the second to last notation.
I'll furthermore introduce the shorthand
So then my general Multivector becomes
Although
The wedge product
The Geometric Product of two Multivectors
Then there is the Flip operation which simply switches the sign of the vector part:
and the Conjugate, which switches the sign of the imagnary part:
Ok, with all this we can finally define inversion:
For this last fraction, the denominator is indeed real-valued. The given inverse is both a left- and a right-inverse, so no special care has to be taken for the order.
I.e.
Finally, here is the result expanded in full component form (no longer with real denominator, but as said, you can treat
Obviously, there are some problems with multivectors for which
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on April 05, 2015, 03:54:09 PM
The Geometric Algebra Homepage at Cambridge University has a new layout. It worth visiting it!
http://geometry.mrao.cam.ac.uk/ (http://geometry.mrao.cam.ac.uk/)
Sadly, I personaly have not much time to work further on geometric algebra but have bought the book Geometric Algebra for Physicsts from Doran/Lasenby and stated reading it.
Being familiar with physics and basics of geometric algebra it is worth reading it!
Hermann
http://geometry.mrao.cam.ac.uk/ (http://geometry.mrao.cam.ac.uk/)
Sadly, I personaly have not much time to work further on geometric algebra but have bought the book Geometric Algebra for Physicsts from Doran/Lasenby and stated reading it.
Being familiar with physics and basics of geometric algebra it is worth reading it!
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on April 09, 2015, 06:02:41 PM
I found another wikipedia page which I think wasn't mentioned in this thread before
https://en.wikipedia.org/wiki/Plane_of_rotation
This one isn't about geometric algebra per se but rather about the concept of rotation in arbitrary numbers of dimensions (featuring the iconic rotating hypercube). It does, however, illustrate how geometric algebra is very useful in describing the notion of rotation and gives a different angle on bivectors (introducing them as rotation rather than "twodimensional" vectors that later are identified as rotations)
https://en.wikipedia.org/wiki/Plane_of_rotation
This one isn't about geometric algebra per se but rather about the concept of rotation in arbitrary numbers of dimensions (featuring the iconic rotating hypercube). It does, however, illustrate how geometric algebra is very useful in describing the notion of rotation and gives a different angle on bivectors (introducing them as rotation rather than "twodimensional" vectors that later are identified as rotations)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: flexiverse on April 10, 2015, 02:21:08 AM
Very interesting this. Having just learnt how important grassmanian algebra is.
What is the best resource to Learn clifford algebra for absolute beginners ?
I always thought there should be a more unified algebra that works in all dimensions! I'm completely mind blown grassmann was doing this so long ago !!!
What is the best resource to Learn clifford algebra for absolute beginners ?
I always thought there should be a more unified algebra that works in all dimensions! I'm completely mind blown grassmann was doing this so long ago !!!
Title: Re: Geometric Algebra, Geometric Calculus
Post by: flexiverse on April 10, 2015, 02:30:03 AM
Actually it just came back to mind. I remember setting Laz Plaths, aka qquazxxsw on YouTube , trochoid applications as my favourite visualisation tools! You can set up an n dimensional trochoid very easly and intuitively on his apps, because each rotating circle represents a dimension.
Check out his channel and videos. Count how many circles he uses!
I honestly think this man has solved the problem of visualising n dimensions in every practicable way. But he won't talk to me!
Oh well, that's life I suppose :D
Er, I can't find qquazxxsw on you tube ?Check out his channel and videos. Count how many circles he uses!
I honestly think this man has solved the problem of visualising n dimensions in every practicable way. But he won't talk to me!
Oh well, that's life I suppose :D
Title: Re: Geometric Algebra, Geometric Calculus
Post by: flexiverse on April 10, 2015, 03:07:16 AM
Very interesting this. Having just learnt how important grassmanian algebra is.
What is the best resource to Learn clifford algebra for absolute beginners ?
I always thought there should be a more unified algebra that works in all dimensions! I'm completely mind blown grassmann was doing this so long ago !!!
What is the best resource to Learn clifford algebra for absolute beginners ?
I always thought there should be a more unified algebra that works in all dimensions! I'm completely mind blown grassmann was doing this so long ago !!!
Just answered my own question !!!
Geometric Algebra for computer graphics book :
http://www.itpa.lt/~acus/Knygos/Clifford_algebra_books/%5BJohn_A._Vince%5D_Geometric_algebra_for_computer_gra%28BookFi.org%29.pdf
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on April 10, 2015, 10:34:10 AM
flexiverse, it may be a tall task by now (considering this thread is 12 pages long already!) but there are a LOT of resources of varying difficulty strewn out all across it. Tendentially the more beginner-level ones are earlier in the thread (although I think I remember there being a couple easier ones in the more recent past as well.)
Though even most of the more technical papers tend to recap the same ideas over and over. They do this because Geometric Algebra isn't in the mainstream of maths and thus, to expand the audience, it needs to always be re-explained.
Thanks for that link though! I may be wrong but I think that one's new to this thread.
In fact this whole thing has some nice things http://www.itpa.lt/~acus/Knygos/Clifford_algebra_books/
Though even most of the more technical papers tend to recap the same ideas over and over. They do this because Geometric Algebra isn't in the mainstream of maths and thus, to expand the audience, it needs to always be re-explained.
Thanks for that link though! I may be wrong but I think that one's new to this thread.
In fact this whole thing has some nice things http://www.itpa.lt/~acus/Knygos/Clifford_algebra_books/
Title: Re: Geometric Algebra, Geometric Calculus
Post by: eiffie on April 10, 2015, 04:13:15 PM
Sweet! Thanks for the link(s).
Title: Re: Geometric Algebra, Geometric Calculus
Post by: flexiverse on April 10, 2015, 07:24:21 PM
flexiverse, it may be a tall task by now (considering this thread is 12 pages long already!) but there are a LOT of resources of varying difficulty strewn out all across it. Tendentially the more beginner-level ones are earlier in the thread (although I think I remember there being a couple easier ones in the more recent past as well.)
Though even most of the more technical papers tend to recap the same ideas over and over. They do this because Geometric Algebra isn't in the mainstream of maths and thus, to expand the audience, it needs to always be re-explained.
Thanks for that link though! I may be wrong but I think that one's new to this thread.
In fact this whole thing has some nice things http://www.itpa.lt/~acus/Knygos/Clifford_algebra_books/
Though even most of the more technical papers tend to recap the same ideas over and over. They do this because Geometric Algebra isn't in the mainstream of maths and thus, to expand the audience, it needs to always be re-explained.
Thanks for that link though! I may be wrong but I think that one's new to this thread.
In fact this whole thing has some nice things http://www.itpa.lt/~acus/Knygos/Clifford_algebra_books/
Mind you I was specifically referring to the book "geometric algebra for computer graphics" by John vince.
But it turns out the root folder is accessible and contains EVERY book and paper about this subject on this entire planet !!! Really !
Fantastic find I must say so my self. O0
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on April 10, 2015, 10:04:34 PM
My Starting point was the paper of Richard Alan Miller. You can find a link in this thread or you can find the link at my Geometric Algebra page, which is inspired by this thread here in fractal forums.
Hermann
http://www.wackerart.de/mathematik/geometric_algebra.html (http://www.wackerart.de/mathematik/geometric_algebra.html)
Hermann
http://www.wackerart.de/mathematik/geometric_algebra.html (http://www.wackerart.de/mathematik/geometric_algebra.html)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: flexiverse on April 10, 2015, 10:28:20 PM
My Starting point was the paper of Richard Alan Miller. You can find a link in this thread or you can find the link at my Geometric Algebra page, which is inspired by this thread here in fractal forums.
Hermann
http://www.wackerart.de/mathematik/geometric_algebra.html (http://www.wackerart.de/mathematik/geometric_algebra.html)
Hermann
http://www.wackerart.de/mathematik/geometric_algebra.html (http://www.wackerart.de/mathematik/geometric_algebra.html)
Thanks. Actually David Hestenes who is mentioned in the first sentence in the introduction in that paper, has produced what looks like the best book on the subject. Namely geometric Algebra for physics.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: jehovajah on April 11, 2015, 11:59:43 PM
Er, I can't find qquazxxsw on you tube ?
Thanks for the correction flexiverse . I have corrected my original post. Here is a link to qqazxxsw for you.
http://m.youtube.com/watch?v=rz8A5l_yn34
Title: Re: Geometric Algebra, Geometric Calculus
Post by: jehovajah on April 12, 2015, 07:22:30 AM
Mind you I was specifically referring to the book "geometric algebra for computer graphics" by John vince.
But it turns out the root folder is accessible and contains EVERY book and paper about this subject on this entire planet !!! Really !
Fantastic find I must say so my self. O0
;D Really? :dink:But it turns out the root folder is accessible and contains EVERY book and paper about this subject on this entire planet !!! Really !
Fantastic find I must say so my self. O0
It is a great resource though, but worth downloading what you want now before the access is controlled more vigorously!
Title: Re: Geometric Algebra, Geometric Calculus
Post by: flexiverse on April 13, 2015, 02:32:40 AM
The john vince books seem great !
Geometric Algebra: An Algebraic System for Computer Games and Animation. By John vince
_Geometric_Algebra_An_Algebraic_Sy(BookFi.org).pdf]http://dl.lux.bookfi.org/genesis/122000/bac7a55b79a54365ad875863c1225bc4/_as/[John_A._Vince]_Geometric_Algebra_An_Algebraic_Sy(BookFi.org).pdf]http://dl.lux.bookfi.org/genesis/122000/bac7a55b79a54365ad875863c1225bc4/_as/[John_A._Vince]_Geometric_Algebra_An_Algebraic_Sy(BookFi.org).pdf]http://dl.lux.bookfi.org/genesis/122000/bac7a55b79a54365ad875863c1225bc4/_as/[John_A._Vince]_Geometric_Algebra_An_Algebraic_Sy(BookFi.org).pdf (http://dl.lux.bookfi.org/genesis/122000/bac7a55b79a54365ad875863c1225bc4/_as/[John_A._Vince)
Honestly guy, geometric algebra is the most fascinating thing I've ever come across!
I've been looking for something like this for a long time, it's the true mathematics, I always knew geometry was the basis for everything.....
I honestly can't believe I've never heard of clifford / grassmann algebra only until recently, it really does unify everything.
Geometric Algebra: An Algebraic System for Computer Games and Animation. By John vince
_Geometric_Algebra_An_Algebraic_Sy(BookFi.org).pdf]http://dl.lux.bookfi.org/genesis/122000/bac7a55b79a54365ad875863c1225bc4/_as/[John_A._Vince]_Geometric_Algebra_An_Algebraic_Sy(BookFi.org).pdf]http://dl.lux.bookfi.org/genesis/122000/bac7a55b79a54365ad875863c1225bc4/_as/[John_A._Vince]_Geometric_Algebra_An_Algebraic_Sy(BookFi.org).pdf]http://dl.lux.bookfi.org/genesis/122000/bac7a55b79a54365ad875863c1225bc4/_as/[John_A._Vince]_Geometric_Algebra_An_Algebraic_Sy(BookFi.org).pdf (http://dl.lux.bookfi.org/genesis/122000/bac7a55b79a54365ad875863c1225bc4/_as/[John_A._Vince)
Honestly guy, geometric algebra is the most fascinating thing I've ever come across!
I've been looking for something like this for a long time, it's the true mathematics, I always knew geometry was the basis for everything.....
I honestly can't believe I've never heard of clifford / grassmann algebra only until recently, it really does unify everything.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on April 13, 2015, 09:47:04 AM
Geometric Algebra can do a lot of awesome things and geometry is very prevalent all throughout both math and life. However, I wouldn't quite go as far as to say it's the basis of everything. A lot of stuff is better described with mere topology which has less structure than geometry. (It's like geometry but without a metric or rather, geometries are topologies equipped with a metric.)
Meanwhile, Topology, Type Theory and Logic can all be combined into a beautifully structured theory called homotopy type theory. It and extensions of it to more general categories seem to be much more natural for a full foundation of mathematics. Geometric Algebra already has too much, too specific structure for that kind of thing.
Though if your sole intention is to manipulate geometric spaces, geometric algebra really is where it's at.
Meanwhile, Topology, Type Theory and Logic can all be combined into a beautifully structured theory called homotopy type theory. It and extensions of it to more general categories seem to be much more natural for a full foundation of mathematics. Geometric Algebra already has too much, too specific structure for that kind of thing.
Though if your sole intention is to manipulate geometric spaces, geometric algebra really is where it's at.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: flexiverse on April 13, 2015, 08:39:37 PM
Geometric Algebra can do a lot of awesome things and geometry is very prevalent all throughout both math and life. However, I wouldn't quite go as far as to say it's the basis of everything. A lot of stuff is better described with mere topology which has less structure than geometry. (It's like geometry but without a metric or rather, geometries are topologies equipped with a metric.)
Meanwhile, Topology, Type Theory and Logic can all be combined into a beautifully structured theory called homotopy type theory. It and extensions of it to more general categories seem to be much more natural for a full foundation of mathematics. Geometric Algebra already has too much, too specific structure for that kind of thing.
Though if your sole intention is to manipulate geometric spaces, geometric algebra really is where it's at.
Meanwhile, Topology, Type Theory and Logic can all be combined into a beautifully structured theory called homotopy type theory. It and extensions of it to more general categories seem to be much more natural for a full foundation of mathematics. Geometric Algebra already has too much, too specific structure for that kind of thing.
Though if your sole intention is to manipulate geometric spaces, geometric algebra really is where it's at.
I' m talking about a much deeper level here. The concept of "sacred geometry" is quite common in the spiritualist realm as the true underlying structure of the universe.
It's only now that physics dudes are seeing how higher dimensional geometry is the real hidden structure if the universe.
E.g. http://www.wired.com/2013/12/amplituhedron-jewel-quantum-physics/ (http://www.wired.com/2013/12/amplituhedron-jewel-quantum-physics/)
E.g. https://www.ted.com/talks/garrett_lisi_on_his_theory_of_everything?language=en
Also the books the approach of geometric algebra in physics clearly state it's obviously the real underlining structure and it should be used because it unifies so much.
Topology is just geometry it's working with higher dimensional objects.
Don't get bogged down with early mathematical thinking that didn't seem to grasp complex numbers and grassmanian alegra,
So that's basically the hidden truth of the universe it's all about higher multi-dimensional objects.
That's why fractals hold a fascination it's working with higher dimensional objects.
In order to explain the universe we need to describe, manipulate higher dimensional objects.
Geometric algebra is just the start of this process, and because of it's clear success in physics just shows that's truly the way the universe works - it's sacred geometry.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on April 13, 2015, 09:12:06 PM
Early mathematical thinking? On the contrary, Homotopy Type Theory is on the very cutting edge of mathematics and newer than Geometric Algebra.
Both of those things unify a lot. Though GA unifies all of geometric manipulation while HoTT unifies three pillars of maths foundation.
You can express GA in HoTT but the opposite will be impossible.
And topology isn't geometry in higher dimensions. Geometry in higher dimensions is geometry in higher dimensions. It's also topology in higher dimensions plus a distance metric.
Both of those things unify a lot. Though GA unifies all of geometric manipulation while HoTT unifies three pillars of maths foundation.
You can express GA in HoTT but the opposite will be impossible.
And topology isn't geometry in higher dimensions. Geometry in higher dimensions is geometry in higher dimensions. It's also topology in higher dimensions plus a distance metric.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: jehovajah on April 23, 2015, 10:33:16 AM
There is a lot to be said for both of your statements, but in fact the more i study Grassmann the more it becomes crucial not to mix up ideas and terminologies and labels.
The Hegelian Philosophy underpins how Grassmann resolved these issues allowing a much broader but precise apprehension of the human experience.
You must "get bogged down" terminologically in each subject division, but you must recognise that you are free to soar above all of them by Analogical thought patterns. What wisdom you as an individual draw from those precise 2 activities is your reward, but personally i cannot impose my apprehension on anyone else. in the same vein i will not accept anyone elses as "absolute".
Yin and Yang are the best approximation i have personal experience of, and of course i am waiting to see how far that pans out! ;D
The Hegelian Philosophy underpins how Grassmann resolved these issues allowing a much broader but precise apprehension of the human experience.
You must "get bogged down" terminologically in each subject division, but you must recognise that you are free to soar above all of them by Analogical thought patterns. What wisdom you as an individual draw from those precise 2 activities is your reward, but personally i cannot impose my apprehension on anyone else. in the same vein i will not accept anyone elses as "absolute".
Yin and Yang are the best approximation i have personal experience of, and of course i am waiting to see how far that pans out! ;D
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on July 20, 2015, 12:05:52 AM
A method to use GA for "holographic reduced representations" or HRRs which are one method to store associations between different terms in vector spaces:
http://arxiv.org/pdf/0710.2611v2.pdf
An example for a HRR would be:
So "France" is defined to be a sum of the "Capital" being "Paris", the "Location" being "WesternEurope" and the "Currency" being "Euro".
Then, if you want to know a fact about France, all you have to do is to multiply with that fact:
This noise is (ideally) relatively small so you can manipulate this result so that you end up with purely Paris.
In the GA variation, this last denoising step becomes especially simple because all the noise ends up in a direction orthogonal to the actual result, so all you have to do is to project the result onto (take the scalar product with) all your plausible terms: For instance, you might know Paris, London and Oslo are Capitals. So you try them all. The largest number resulting from this scalar product will be your answer. A sort of vector based memory.
Also interesting about this paper is how they start explaining GA. I was a little confused at first but they actually chose this interpretation well and it also is an interesting demonstration that the non-commutativity of GA is a direct consequence of it being able to talk about direction. In an undirection version of GA, where = \left( \rightarrow \right) = \left( - \right))
, you'll actually end up with something commutative. (Though also something way less useful: It could represent lines and surfaces and the like but it couldn't tell you what direction you go along a given path or what way you turn, etc.)
http://arxiv.org/pdf/0710.2611v2.pdf
An example for a HRR would be:
Code:
France = [Capital*Paris + Location*WesternEurope + Currency*Euro]
Then, if you want to know a fact about France, all you have to do is to multiply with that fact:
Code:
WhatCity = Capital *France
= Capital * [Capital*Paris + Location*WesternEurope + Currency*Euro]
= Capital*Capital*Paris + Capital*Location*WesternEurope + Capital*Currency*Euro
= Paris + (noise terms).
In the GA variation, this last denoising step becomes especially simple because all the noise ends up in a direction orthogonal to the actual result, so all you have to do is to project the result onto (take the scalar product with) all your plausible terms: For instance, you might know Paris, London and Oslo are Capitals. So you try them all. The largest number resulting from this scalar product will be your answer. A sort of vector based memory.
Also interesting about this paper is how they start explaining GA. I was a little confused at first but they actually chose this interpretation well and it also is an interesting demonstration that the non-commutativity of GA is a direct consequence of it being able to talk about direction. In an undirection version of GA, where
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on July 21, 2015, 02:45:17 AM
Geometric Algebra in Quantum Information Processing http://arxiv.org/pdf/quant-ph/0004031v3.pdf and
A Geometric Algebra Perspective On Quantum Computational Gates And Universality In Quantum Computing http://arxiv.org/pdf/1006.2071.pdf
A Geometric Algebra Perspective On Quantum Computational Gates And Universality In Quantum Computing http://arxiv.org/pdf/1006.2071.pdf
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on August 25, 2015, 05:37:42 PM
Cl(8,2) - "Double Conformal Geometric Algebra (or DCGA)" http://www.rxiv.org/pdf/1508.0086v1.pdf
Remember Cl(6,3)? (Qadratic Geometric Algebra or QGA) That was capable of representing all conic sections (but not in arbitrary rotations for certain objects)
Cl(8,2) goes one step further, also being able to represent a torus natively, and all its objects can be rotated arbitrarily (but not all objects can be intersected)
Perhaps allowing all intersections and in arbitrary rotations could be done with Cl(12,6) ("Double Quadratic Geometric Algebra") or Cl(12,3) or Cl(16,4) ("Triple" and "Quadruple Conformal Geometric Algebra) or similar such constructs. People still investigate this.
Perhaps we could even have a very systematic method to tackle)
in a computationally tractable manner, identifying arbitrary subspaces with the simplest possible polynomials that describe them, but that'll take a lot of thinking and analyzing still.
Remember Cl(6,3)? (Qadratic Geometric Algebra or QGA) That was capable of representing all conic sections (but not in arbitrary rotations for certain objects)
Cl(8,2) goes one step further, also being able to represent a torus natively, and all its objects can be rotated arbitrarily (but not all objects can be intersected)
Perhaps allowing all intersections and in arbitrary rotations could be done with Cl(12,6) ("Double Quadratic Geometric Algebra") or Cl(12,3) or Cl(16,4) ("Triple" and "Quadruple Conformal Geometric Algebra) or similar such constructs. People still investigate this.
Perhaps we could even have a very systematic method to tackle
Title: David Hestenes - Tutorial on Geometric Calculus
Post by: hermann on March 19, 2016, 10:39:23 AM
https://www.youtube.com/watch?v=ItGlUbFBFfc (https://www.youtube.com/watch?v=ItGlUbFBFfc)
A fascinating video of David Hestenes on his work with geometric algebra and geometric calculus the relations to physics and other parts of mathematics.
He also describes his truble on getting his importened work published.
Hermann
A fascinating video of David Hestenes on his work with geometric algebra and geometric calculus the relations to physics and other parts of mathematics.
He also describes his truble on getting his importened work published.
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on March 19, 2016, 02:27:18 PM
nice, thanks for sharing! - I wish he'd go a little more in-depth with the actual operations though. This is hardly a "tutorial"
Title: Geometric Algebra for Physicists
Post by: hermann on March 25, 2016, 09:08:47 PM
This thread has always been a great inspriration for me since I first read it the first time about two years ago.
During my physics study I was trained in vector analysis as the mathematics nessesarry for doing physics. It was a special course math for physics.
I was wondering if this is the complete story. I was told that if I had understood all the abstract math of the math courses that ran parallel all would become clear.
But this was not the case. I was thought a lot of abstract math where I often could not find the connection to physics.
Near the end of the math courses I asked the Professor of the relations between his lessons and physics he told me that he was a mathematician an had no deep understanding of physics and that his lesson was for mathematicians only. So I had been tought the wrong mathematics.
Nowdays I agree with the critics of modern mathematics (the mathematics I have been told) in the way that Norman Wildberger stated in some of his lectures:
http://wacker-art.blogspot.de/2016/01/why-infinite-sets-dont-exist.html (http://wacker-art.blogspot.de/2016/01/why-infinite-sets-dont-exist.html)
When I later tried to understand general theory of relativety I also discovered that the required maths had not been tought. I had to work my self through tensors and up and down indizies and was wondering if this had to be such complicate.
When I discovered geometric algebra here in Fractal Forums, this was an eye opener. This was the mathematics I was searching for.
I went through many of the papers posted in this thread that gave me a basic understanding of geometric algebra. I also started a geometric algebra page on my homepage in the internet.http://www.wackerart.de/mathematik/geometric_algebra.html (http://www.wackerart.de/mathematik/geometric_algebra.html)
I then bought my first book on geometric algebra "Geometric Algebra for Physicists - Chris Doran Anthony Lasenby" which is an excellent book.
http://geometry.mrao.cam.ac.uk/2007/01/geometric-algebra-for-physicists/ (http://geometry.mrao.cam.ac.uk/2007/01/geometric-algebra-for-physicists/)
As the title says it is an book for physicists and I have read some of the basic papers referenced in this thread. My favorit paper is still the one of Richard Allan Miller.
But with this preperation this is the right book to start digging deeper into physics. Many basic physics equations reveal more of there structure when written in the language of geometric algebra. So geometric algebra gives a clearer and easier access to physics, as if you know only the language of Gibb's Vector analysis.
This book also gave me a deeper understanding of geometric algebra.
Some of the chapters I have read several times and always discover something new.
Hermann
During my physics study I was trained in vector analysis as the mathematics nessesarry for doing physics. It was a special course math for physics.
I was wondering if this is the complete story. I was told that if I had understood all the abstract math of the math courses that ran parallel all would become clear.
But this was not the case. I was thought a lot of abstract math where I often could not find the connection to physics.
Near the end of the math courses I asked the Professor of the relations between his lessons and physics he told me that he was a mathematician an had no deep understanding of physics and that his lesson was for mathematicians only. So I had been tought the wrong mathematics.
Nowdays I agree with the critics of modern mathematics (the mathematics I have been told) in the way that Norman Wildberger stated in some of his lectures:
http://wacker-art.blogspot.de/2016/01/why-infinite-sets-dont-exist.html (http://wacker-art.blogspot.de/2016/01/why-infinite-sets-dont-exist.html)
When I later tried to understand general theory of relativety I also discovered that the required maths had not been tought. I had to work my self through tensors and up and down indizies and was wondering if this had to be such complicate.
When I discovered geometric algebra here in Fractal Forums, this was an eye opener. This was the mathematics I was searching for.
I went through many of the papers posted in this thread that gave me a basic understanding of geometric algebra. I also started a geometric algebra page on my homepage in the internet.http://www.wackerart.de/mathematik/geometric_algebra.html (http://www.wackerart.de/mathematik/geometric_algebra.html)
I then bought my first book on geometric algebra "Geometric Algebra for Physicists - Chris Doran Anthony Lasenby" which is an excellent book.
http://geometry.mrao.cam.ac.uk/2007/01/geometric-algebra-for-physicists/ (http://geometry.mrao.cam.ac.uk/2007/01/geometric-algebra-for-physicists/)
As the title says it is an book for physicists and I have read some of the basic papers referenced in this thread. My favorit paper is still the one of Richard Allan Miller.
But with this preperation this is the right book to start digging deeper into physics. Many basic physics equations reveal more of there structure when written in the language of geometric algebra. So geometric algebra gives a clearer and easier access to physics, as if you know only the language of Gibb's Vector analysis.
This book also gave me a deeper understanding of geometric algebra.
Some of the chapters I have read several times and always discover something new.
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on March 26, 2016, 01:32:37 AM
Thanks for bringing up this website again!
Looks like Chris Dolan was actually holding a course on Geometric Algebra last year. - It was linked right there on the page where the book is promoted.
http://geometry.mrao.cam.ac.uk/2015/10/geometric-algebra-2015/
Each of those lectures also include slides. I'd imagine they match roughly with the book but I don't know: I haven't read it.
Looks like Chris Dolan was actually holding a course on Geometric Algebra last year. - It was linked right there on the page where the book is promoted.
http://geometry.mrao.cam.ac.uk/2015/10/geometric-algebra-2015/
Each of those lectures also include slides. I'd imagine they match roughly with the book but I don't know: I haven't read it.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on March 26, 2016, 06:39:49 AM
The slides of the lectures are close to the formulas in the book, so they give a good overview or may be used as a formular collection. But the book contains very much descriptive text, that makes it all understandable. The book covers much more topics then the slices of the lessons that are available. The lecture is mainly on the geometric algebra part of the book. Most of the physiks is not covered.
Hermann
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on March 26, 2016, 12:00:58 PM
Well yeah, hopefully the book covers more than that :)
Found a new paper. This one is super abstract though. It's about how to extend Conformal Geometric Algebra to all kinds of spaces. - That is, ones over something other than real numbers. It'll work with
So what they are saying is that for integers modulo 2 only parts of the construction work, but for integers modulo 3 and any field larger than that, this should work fine.
Some interesting infinitely large structures that should become possible with this are things like p-adic numbers, I think. Not entirely sure.
Here is the link: http://arxiv.org/pdf/1603.06863.pdf
Found a new paper. This one is super abstract though. It's about how to extend Conformal Geometric Algebra to all kinds of spaces. - That is, ones over something other than real numbers. It'll work with
Quote
any field where the characteristic is not 2, and some of the constructions work even in that case.
A field of characteristic 2 is, for instance, something like integers modulo 2. If you multiply the multiplicative identity (1) with 2, in the integers modulo 2, you'll get back the additive identity (0).So what they are saying is that for integers modulo 2 only parts of the construction work, but for integers modulo 3 and any field larger than that, this should work fine.
Some interesting infinitely large structures that should become possible with this are things like p-adic numbers, I think. Not entirely sure.
Here is the link: http://arxiv.org/pdf/1603.06863.pdf
Title: Re: David Hestenes - Tutorial on Geometric Calculus
Post by: jehovajah on March 31, 2016, 12:55:56 AM
https://m.youtube.com/watch?v=ItGlUbFBFfc (https://m.youtube.com/watch?v=ItGlUbFBFfc)
A fascinating video of David Hestenes on his work with geometric algebra and geometric calculus the relations to physics and other parts of mathematics.
He also describes his truble on getting his importened work published.
Hermann
For older platforms .A fascinating video of David Hestenes on his work with geometric algebra and geometric calculus the relations to physics and other parts of mathematics.
He also describes his truble on getting his importened work published.
Hermann
Title: Caylec_Dickson Construction of Hyper Complex Numbers
Post by: hermann on April 03, 2016, 02:19:55 PM
How to build Hypercomplex-Numbers from normal complex Numbers is done by the Cayley-Dickson Construction:
For details look at the following page John Baez:
http://math.ucr.edu/home/baez/octonions/node5.html (http://math.ucr.edu/home/baez/octonions/node5.html)
There is also an articel in Wikipedia:
https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction (https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction)
The History of Hyper Complex Numbers
http://history.hyperjeff.net/hypercomplex (http://history.hyperjeff.net/hypercomplex)
Hermann
For details look at the following page John Baez:
http://math.ucr.edu/home/baez/octonions/node5.html (http://math.ucr.edu/home/baez/octonions/node5.html)
There is also an articel in Wikipedia:
https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction (https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction)
The History of Hyper Complex Numbers
http://history.hyperjeff.net/hypercomplex (http://history.hyperjeff.net/hypercomplex)
Hermann
Title: Video Geometric Calculus - Alan Macdonald
Post by: hermann on September 15, 2016, 07:33:11 AM
I have just watched the following video series on Geometric Calculus by Alan Macdonald.
It gives a very good Introduction to the basics of Geometric Calculus.
https://www.youtube.com/watch?v=-JQxOYL3vhY (https://www.youtube.com/watch?v=-JQxOYL3vhY)
An eye opener for people with basic knowledge on geometric algebra, vector analysis and complex analysis.
For me many of the things presented in this video series are well known.
But this videos shows very good how all works together.
Hermann
It gives a very good Introduction to the basics of Geometric Calculus.
https://www.youtube.com/watch?v=-JQxOYL3vhY (https://www.youtube.com/watch?v=-JQxOYL3vhY)
An eye opener for people with basic knowledge on geometric algebra, vector analysis and complex analysis.
For me many of the things presented in this video series are well known.
But this videos shows very good how all works together.
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 15, 2016, 01:15:21 PM
That's actually a video series or two, one about GA, one about GC.
I'm glad these exist but I wish his style would be a little less "read-off-the-slides-y" and maybe more "hands on" if that makes sense?
I'm glad these exist but I wish his style would be a little less "read-off-the-slides-y" and maybe more "hands on" if that makes sense?
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on September 15, 2016, 06:43:38 PM
Hallo Kram! I have changed it from video to video series. Have you watched the whole series?
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 15, 2016, 06:52:09 PM
yes, my comment applied to the series as a whole, not to just that one video :)
It's not a long series, so it's a fairly quick watch.
It's not a long series, so it's a fairly quick watch.
Title: Re: Video Geometric Calculus - Alan Macdonald
Post by: jehovajah on March 01, 2017, 10:29:56 AM
I have just watched the following video series on Geometric Calculus by Alan Macdonald.
It gives a very good Introduction to the basics of Geometric Calculus.
https://m.youtube.com/watch?v=-JQxOYL3vhY (https://m.youtube.com/watch?v=-JQxOYL3vhY)
An eye opener for people with basic knowledge on geometric algebra, vector analysis and complex analysis.
For me many of the things presented in this video series are well known.
But this videos shows very good how all works together.
Hermann
It gives a very good Introduction to the basics of Geometric Calculus.
https://m.youtube.com/watch?v=-JQxOYL3vhY (https://m.youtube.com/watch?v=-JQxOYL3vhY)
An eye opener for people with basic knowledge on geometric algebra, vector analysis and complex analysis.
For me many of the things presented in this video series are well known.
But this videos shows very good how all works together.
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: macawscopes on March 31, 2017, 12:30:52 AM
Apparently there are generalized 'Clifford algebras' that can be constructed from higher degree homogeneous polynomials. From what I can tell, these generalized 'Clifford algebras' have natural ternary, quartic, etc. operations instead of binary products.
Ordinary Clifford algebras can be made by starting with the tensor algebra over a vector space, then sculpting it down using equivalence relations made between bivectors elements of the tensor algebra and corresponding scalars of a quadratic form.
Say
is a Clifford algebra of the quadratic form 
, then it has a binary product (a.k.a. the geometric product) 
. The quadratic form is used to take the squares of grade-1 elements and equate them with scalars, e.g. 
, etc. This is what trims down the geometric product and keeps it closed.
So... you can actually do something similar using algebraic forms, which are a generalization of quadratic forms. A quadratic form is linked to a homogeneous quadratic polynomial. Algebraic forms are linked to homogeneous polynomials (http://en.wikipedia.org/wiki/Homogeneous_polynomial) of any degree, e.g. cubic forms (http://en.wikipedia.org/wiki/Cubic_form)
A homogeneous polynomial
of degree 
in 
variables, can be thought of as a function  = 0)
. This can be used in an analogous way to generate equivalence relations such as 
. I've only seen papers about ternary ones,but I think theoretically you could create an n-ary algebra analogous to a Clifford Algebra from any homogeneous polynomial.
According to this paper (http://arxiv.org/abs/math-ph/0011023), you cannot necessarily compose the ternary operation of a ternary algebra from repeated application of some binary operation. So these things are fundamentally different from the familiar binary algebras.
I'm still scratching my head on how to actually formulate these things. But it's starting to make more sense as time goes on....
One clue is here (http://en.wikipedia.org/wiki/Symmetric_tensor). A symmetric tensor is the analog of a symmetric matrix, and it is decomposable in an analogous way to how symmetric matrices are diagonalizable. Wikipedia says that: "The space of symmetric tensors of order r on a finite-dimensional vector space is naturally isomorphic to the dual of the space of homogeneous polynomials of degree r on V"... which I translate to mean that for every homogeneous polynomial of degree r on V, there is a unique symmetric tensor of order r.
Here is the scoop on decomposition of symmetric tensors (http://en.wikipedia.org/wiki/Symmetric_tensor#Decomposition)
Do a search for "ternary clifford algebra" and see what you come up with. There are quite a few papers on it, though most of them are really, really hard to understand.
Obviously, I'd love to make fractals out of these things!
Ordinary Clifford algebras can be made by starting with the tensor algebra over a vector space, then sculpting it down using equivalence relations made between bivectors elements of the tensor algebra and corresponding scalars of a quadratic form.
Say
So... you can actually do something similar using algebraic forms, which are a generalization of quadratic forms. A quadratic form is linked to a homogeneous quadratic polynomial. Algebraic forms are linked to homogeneous polynomials (http://en.wikipedia.org/wiki/Homogeneous_polynomial) of any degree, e.g. cubic forms (http://en.wikipedia.org/wiki/Cubic_form)
A homogeneous polynomial
According to this paper (http://arxiv.org/abs/math-ph/0011023), you cannot necessarily compose the ternary operation of a ternary algebra from repeated application of some binary operation. So these things are fundamentally different from the familiar binary algebras.
I'm still scratching my head on how to actually formulate these things. But it's starting to make more sense as time goes on....
One clue is here (http://en.wikipedia.org/wiki/Symmetric_tensor). A symmetric tensor is the analog of a symmetric matrix, and it is decomposable in an analogous way to how symmetric matrices are diagonalizable. Wikipedia says that: "The space of symmetric tensors of order r on a finite-dimensional vector space is naturally isomorphic to the dual of the space of homogeneous polynomials of degree r on V"... which I translate to mean that for every homogeneous polynomial of degree r on V, there is a unique symmetric tensor of order r.
Here is the scoop on decomposition of symmetric tensors (http://en.wikipedia.org/wiki/Symmetric_tensor#Decomposition)
Do a search for "ternary clifford algebra" and see what you come up with. There are quite a few papers on it, though most of them are really, really hard to understand.
Obviously, I'd love to make fractals out of these things!
Title: Re: Geometric Algebra, Geometric Calculus
Post by: macawscopes on March 31, 2017, 02:35:47 AM
By the way... here's another great resource on quadratic forms and homogeneous polynomials (the fundamental underpinnings of Clifford Algebras) http://www.cs.du.edu/~petr/data/papers/symmetric_multilinear_forms_and_polarization_of_polynomials.pdf (http://www.cs.du.edu/~petr/data/papers/symmetric_multilinear_forms_and_polarization_of_polynomials.pdf)
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on April 16, 2017, 10:48:22 AM
Hallo Macawscopes,
thanks for your contributions to this thread.
But on the first view I can not see the connection to geometric algebra and geometric calculus.
Your contributions are very abstract and for me difficult to read.
Most of your links don't work! Please check them all.
Do you have a paper that gives an easy introduction like the paper of
Richard Alan Miller on Geometric Algebra?
Hermann
thanks for your contributions to this thread.
But on the first view I can not see the connection to geometric algebra and geometric calculus.
Your contributions are very abstract and for me difficult to read.
Most of your links don't work! Please check them all.
Do you have a paper that gives an easy introduction like the paper of
Richard Alan Miller on Geometric Algebra?
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: macawscopes on April 17, 2017, 03:53:34 AM
Thanks for the tip on the links, Hermann. I fixed them!
Unfortunately, I don't think there are such straightforward introductions to this kind of thing (general Clifford Algebras), as nobody like David Hestenes has come along to interpret them pedagogically in the same way as Hestenes has done for the Clifford algebras of quadratic forms... as far as I know.
Basically, geometric algebra is a way of studying Clifford algebra, but Clifford algebra is a much broader subject, related to creating algebras from a basic mathematical object called a homogeneous polynomial. A quadratic form is a kind of homogeneous polynomial, the kind that geometric algebra focuses on. But there are many kinds of homogeneous polynomials.
Homogeneous polynomials: Just choose some variables, say x and y, then choose a degree. A quadratic form would be degree 2.
is a homogenous polynomial of degree 2. Here's a homogeneous polynomial of degree 4: 
.
Clifford algebra is about is using homogeneous polynomials to come up with particular algebraic cancellations (/equivalencies/rules) that define an algebra. Each polynomial defines its own set of algebraic rules, and therefore its own characteristic algebra.
From what I can tell, the rules derived from homogeneous polynomials of degree 3 (i.e.
) have a natural interpretation as rules of a ternary operation. So imagine if a geometric product took 3 arguments instead of 2.
This is the takeaway: given a homogeneous polynomial, you can come up with a Clifford algebra with its rules derived from that polynomial. Its characteristic operation doesn't have to be a binary 'product'. It could be 3-ary, 4-ary, etc. Geometric algebra is the study of Clifford algebras with 2-ary operations, derived from quadratic forms (a.k.a. homogeneous polynomials of degree 2). But the world of Clifford algebra is much, much bigger!
Unfortunately, I don't think there are such straightforward introductions to this kind of thing (general Clifford Algebras), as nobody like David Hestenes has come along to interpret them pedagogically in the same way as Hestenes has done for the Clifford algebras of quadratic forms... as far as I know.
Basically, geometric algebra is a way of studying Clifford algebra, but Clifford algebra is a much broader subject, related to creating algebras from a basic mathematical object called a homogeneous polynomial. A quadratic form is a kind of homogeneous polynomial, the kind that geometric algebra focuses on. But there are many kinds of homogeneous polynomials.
Homogeneous polynomials: Just choose some variables, say x and y, then choose a degree. A quadratic form would be degree 2.
Clifford algebra is about is using homogeneous polynomials to come up with particular algebraic cancellations (/equivalencies/rules) that define an algebra. Each polynomial defines its own set of algebraic rules, and therefore its own characteristic algebra.
From what I can tell, the rules derived from homogeneous polynomials of degree 3 (i.e.
This is the takeaway: given a homogeneous polynomial, you can come up with a Clifford algebra with its rules derived from that polynomial. Its characteristic operation doesn't have to be a binary 'product'. It could be 3-ary, 4-ary, etc. Geometric algebra is the study of Clifford algebras with 2-ary operations, derived from quadratic forms (a.k.a. homogeneous polynomials of degree 2). But the world of Clifford algebra is much, much bigger!
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on April 17, 2017, 07:11:50 PM
But it is possible to build the geometric product of three vectors, so what is new?
Title: Re: Geometric Algebra, Geometric Calculus
Post by: macawscopes on April 17, 2017, 10:31:19 PM
From what I've read, some of these ternary operators can't be decomposed into repeated application of associative binary products. So in other words, there's something fundamentally different about these operators... they aren't just simple compositions of binary operations...
But really, I'm very much in the same boat as you as far as my understanding goes. My limited understanding is coming from that I've been persistent in digging into a lot of papers about this, but it's still very hard to understand! I've been tempted to email some authors of these papers to get a better layman's understanding of how the different perspectives relate.
But really, I'm very much in the same boat as you as far as my understanding goes. My limited understanding is coming from that I've been persistent in digging into a lot of papers about this, but it's still very hard to understand! I've been tempted to email some authors of these papers to get a better layman's understanding of how the different perspectives relate.
Title: Geometric Algebra, Geometric Calculus
Post by: hermann on April 29, 2017, 11:02:25 AM
Hallo Macawscopes,
Octonion form a nonassociative algebra.
On Octonians there is a nice video on You Tube from John Baez:
https://www.youtube.com/watch?v=Tw8w4YPp4zM (https://www.youtube.com/watch?v=Tw8w4YPp4zM)
I also found some information on Spinors and Trialities on the hompage of John Baez.
http://math.ucr.edu/home/baez/octonions/node7.html (http://math.ucr.edu/home/baez/octonions/node7.html)
May be this helps a bit.
Hermann
Octonion form a nonassociative algebra.
On Octonians there is a nice video on You Tube from John Baez:
https://www.youtube.com/watch?v=Tw8w4YPp4zM (https://www.youtube.com/watch?v=Tw8w4YPp4zM)
I also found some information on Spinors and Trialities on the hompage of John Baez.
http://math.ucr.edu/home/baez/octonions/node7.html (http://math.ucr.edu/home/baez/octonions/node7.html)
May be this helps a bit.
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: jehovajah on May 01, 2017, 12:02:30 PM
Xxx Hermann Grassmann made a simple point: when the symbolic manipulations become complex and not geometrically or naturally intepretable there is a big incentive not to pursue the algebra too deeply.
So the Cotes DeMoivre roots of unity theory were pursuable because they were relatable to the arcs on a circle. Similarly Quaternions are relatable to spherical triangular surfaces etc.
The higher homogenous forms, see Norman Wildberger about universal hyperbolic geometry, have practical pplications or they don't warrant a great deal of exploration .
However, my recent exploration into trochoidal surfaces makes the case for using these higher homogenous forms to model spectral data and dynamics in the electromagnetic spectrum .
Quite a number of my Quasz fractal sculptures hint at correlations between these frequency measurements and perceived forms in materiality.
So the Cotes DeMoivre roots of unity theory were pursuable because they were relatable to the arcs on a circle. Similarly Quaternions are relatable to spherical triangular surfaces etc.
The higher homogenous forms, see Norman Wildberger about universal hyperbolic geometry, have practical pplications or they don't warrant a great deal of exploration .
However, my recent exploration into trochoidal surfaces makes the case for using these higher homogenous forms to model spectral data and dynamics in the electromagnetic spectrum .
Quite a number of my Quasz fractal sculptures hint at correlations between these frequency measurements and perceived forms in materiality.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on May 01, 2017, 02:21:34 PM
Thanks Jehovajah,
I only wanted to give macawscopes some hints where he can look for what he is searching for. The Homepage of John Baez is the closesd I could find.
I also rember a discussion with Kram in this thread on the relation between Oktonion, Septonions etc. Which is not so direct as we originaly expected.
Some hints on this issue can also be found on John Baez pages on Clifford Algebra.
I also remebered that I have a Book called "A New Approach to Differential Geometry using Cliffords's Geometric Algebra" by John Snygg.
I revisited the book but also could not find hints on what macawscopes is asking for.
(But started reading some Chapters I haven't read and enjoyed it.)
Hermann
I only wanted to give macawscopes some hints where he can look for what he is searching for. The Homepage of John Baez is the closesd I could find.
I also rember a discussion with Kram in this thread on the relation between Oktonion, Septonions etc. Which is not so direct as we originaly expected.
Some hints on this issue can also be found on John Baez pages on Clifford Algebra.
I also remebered that I have a Book called "A New Approach to Differential Geometry using Cliffords's Geometric Algebra" by John Snygg.
I revisited the book but also could not find hints on what macawscopes is asking for.
(But started reading some Chapters I haven't read and enjoyed it.)
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on May 09, 2017, 11:49:40 AM
An Interview with David Hestenes:
His life and achievements
http://www.ejmste.com/v8n2/eurasia_v8n2_tasar.pdf (http://www.ejmste.com/v8n2/eurasia_v8n2_tasar.pdf)
Very intresting are his talks on Zitterbewegung and his Interpretation of Schrödinger and Dirac equations.
Hermann
His life and achievements
http://www.ejmste.com/v8n2/eurasia_v8n2_tasar.pdf (http://www.ejmste.com/v8n2/eurasia_v8n2_tasar.pdf)
Very intresting are his talks on Zitterbewegung and his Interpretation of Schrödinger and Dirac equations.
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on May 21, 2017, 09:04:51 PM
Hallo macawscopes,
the last days I have worked out some basics of the Clifford Algebra starting with the geometric product of two vectors.
Have a look on my geometric algebra in the Cliffor Algebra section page.
http://www.wackerart.de/mathematik/geometric_algebra.html#clifford (http://www.wackerart.de/mathematik/geometric_algebra.html)
It should be possible to expand this to higher order blades.
May be the helps a little bit to decipher some of the abstract papers you have posted.
Hermann
the last days I have worked out some basics of the Clifford Algebra starting with the geometric product of two vectors.
Have a look on my geometric algebra in the Cliffor Algebra section page.
http://www.wackerart.de/mathematik/geometric_algebra.html#clifford (http://www.wackerart.de/mathematik/geometric_algebra.html)
It should be possible to expand this to higher order blades.
May be the helps a little bit to decipher some of the abstract papers you have posted.
Hermann
Title: The Trible Quad Formular
Post by: hermann on July 08, 2017, 03:42:32 PM
The following lesson from Norman Wildberger may be helpfull for Macawscopes Question on trible products:
https://www.youtube.com/watch?v=e17J0mOhelQ (https://www.youtube.com/watch?v=e17J0mOhelQ)
It is intresting to compare some of Norman Wildbergers formulars with the formulars on Clifford Algebra I have derivated from the geometric product on my geometric algebra page.
http://www.wackerart.de/mathematik/geometric_algebra.html#clifford (http://www.wackerart.de/mathematik/geometric_algebra.html#clifford)
Hermann
https://www.youtube.com/watch?v=e17J0mOhelQ (https://www.youtube.com/watch?v=e17J0mOhelQ)
It is intresting to compare some of Norman Wildbergers formulars with the formulars on Clifford Algebra I have derivated from the geometric product on my geometric algebra page.
http://www.wackerart.de/mathematik/geometric_algebra.html#clifford (http://www.wackerart.de/mathematik/geometric_algebra.html#clifford)
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: jehovajah on July 17, 2017, 05:48:48 AM
Cambridge University published “Geometric Algebra for Physicists.” That book arose from more than a decade of GA research at Cambridge that produced many important results, most notably, “Gauge Theory Gravity,” which improves on General Relativity. Now GA is being applied to robotics and there are conferences on GA every year around the world. It is clear now that the whole field will keep growing without my help. My ultimate goal has always been to see GA become a standard, unified language for physics and engineering as well as mathematics. GA is arguably the optimal mathematical language for physics. For example, you can do introductory physics using geometric algebra without using any coordinates. Actually, my Oersted Medal lecture, published in the American Journal of Physics, is an introduction to geometric algebra at an elementary level. So, I’m willing to bet that GA will eventually become the standard language, even in high school. There is a need to integrate high school algebra, geometry, and trigonometry into one coherent system that is also applicable to physics. GA puts it all together in a remarkable way.
T: So is it easy to make sense for . . .?
H: Well, you see, if you’ve already learned a different language, right? A new language looks hard.
T: Yes.
H: No matter what language! However, if you analyze GA in terms of its structure, it can’t be harder than conventional mathematics, because its assumptions are simpler. The geometric interpretation it gives to algebraic operation is more direct and richer than ordinary vector algebra. It includes all the features of ordinary vector algebra, but it’s not limited to three dimensions. It works in space-time, and so you have a vector algebra for space-time, which, as I have noted already, improves on the Dirac algebra. Indeed, it turns out that I discovered something amazing when I reformulated the Dirac equation in terms of space-time algebra, where Dirac’s gammas –the gamma matrices– are now vectors, okay? The gammas become an orthonormal frame of vectors in space-time. But, what about the imaginary unit i in quantum mechanics? Well, it turns out that you don’t need it.
T: You don’t need it?
H: You don’t need it! You don’t need an extra imaginary unit because the frame of orthonormal vectors suffices when multiplication of vectors is defined by the rules of geometric algebra. Of the four vectors in a frame, one is a timelike vector and three are spacelike vectors, right? If you take the product of two spacelike vectors you get a new quantity called a bivector, which generates rotations in a plane of the two vectors, and its square is minus one. As I proved in 1967 (in the Journal of Mathematical Physics) the generator of phase in the Dirac wave function is just
such a bivector. And what is the physical significance of the plane specified by that bivector? Well, that plane determines the direction of the spin. Thus, spin and complex numbers are intimately, indeed, inseparably related in the Dirac equation. You cannot see that in the ordinary matrix formulation, because the geometry is suppressed. Because matrix algebra is not a geometric algebra; it was developed as a purely formal approach to handle systems of linear equations. In contrast, geometric algebra gives the Dirac equation geometric meaning. So, there is a meaning to the imaginary unit i that appears in the Dirac equation. We have seen that it represents the plane of spin. Eventually, I also proved that this property remains when you do the non- relativistic approximation to the Dirac equation, going to the Pauli equation, and then to the Schrödinger equation. Now, it is usually said that the Schrödinger equation describes a particle without spin. But, the fact is, when you do the approximation correctly this i, which generates rotation in a plane in the Dirac equation, remains precisely as the i in the Schrödinger equation. Thus, the i in the Schrödinger equation is generator of rotations in a plane, and the normal to that plane is a spin direction. In other words, the Schrödinger equation is not describing a particle without spin; it is describes a particle in an eigenstate of spin, that is, with a fixed spin direction. Studying the implications of these facts has been a major theme of my research to this day. And more results will be published soon.
T: Great
H: Yeah, so, that keeps me going. .
T: And you’re still excited after forty years?
H: Yeah, that's right. So, if you are interested I tell you a little about what it has all lead to. Have you heard of zitterbewegung?
T: I’m not familiar.
H: That’s a German word meaning “trembling motion.” The term was coined by Schrödinger. He noticed that if you try to make a wave packet with the free particle solutions of the Dirac equation something funny happens. You can’t make a wave packet using only the positive energy solutions. The Dirac equation has troubles because there are both positive and negative energy solutions, and everybody believes that for a free particle the energy has to be positive. And, you need both positive and negative energy solutions to make wave packets, otherwise you don’t have a complete set. When you make a wave packet it has oscillations between positive and negative states that Schrödinger called zitterbewegung. The frequency of these oscillations is twice the de Broglie frequency. Do you know the de Broglie frequency?
T: Hmm!
H: It is mc squared over h-bar.. The zitterbewegung frequency is twice that, okay? Schrödinger
Thanks for the reference Hermann
T: So is it easy to make sense for . . .?
H: Well, you see, if you’ve already learned a different language, right? A new language looks hard.
T: Yes.
H: No matter what language! However, if you analyze GA in terms of its structure, it can’t be harder than conventional mathematics, because its assumptions are simpler. The geometric interpretation it gives to algebraic operation is more direct and richer than ordinary vector algebra. It includes all the features of ordinary vector algebra, but it’s not limited to three dimensions. It works in space-time, and so you have a vector algebra for space-time, which, as I have noted already, improves on the Dirac algebra. Indeed, it turns out that I discovered something amazing when I reformulated the Dirac equation in terms of space-time algebra, where Dirac’s gammas –the gamma matrices– are now vectors, okay? The gammas become an orthonormal frame of vectors in space-time. But, what about the imaginary unit i in quantum mechanics? Well, it turns out that you don’t need it.
T: You don’t need it?
H: You don’t need it! You don’t need an extra imaginary unit because the frame of orthonormal vectors suffices when multiplication of vectors is defined by the rules of geometric algebra. Of the four vectors in a frame, one is a timelike vector and three are spacelike vectors, right? If you take the product of two spacelike vectors you get a new quantity called a bivector, which generates rotations in a plane of the two vectors, and its square is minus one. As I proved in 1967 (in the Journal of Mathematical Physics) the generator of phase in the Dirac wave function is just
such a bivector. And what is the physical significance of the plane specified by that bivector? Well, that plane determines the direction of the spin. Thus, spin and complex numbers are intimately, indeed, inseparably related in the Dirac equation. You cannot see that in the ordinary matrix formulation, because the geometry is suppressed. Because matrix algebra is not a geometric algebra; it was developed as a purely formal approach to handle systems of linear equations. In contrast, geometric algebra gives the Dirac equation geometric meaning. So, there is a meaning to the imaginary unit i that appears in the Dirac equation. We have seen that it represents the plane of spin. Eventually, I also proved that this property remains when you do the non- relativistic approximation to the Dirac equation, going to the Pauli equation, and then to the Schrödinger equation. Now, it is usually said that the Schrödinger equation describes a particle without spin. But, the fact is, when you do the approximation correctly this i, which generates rotation in a plane in the Dirac equation, remains precisely as the i in the Schrödinger equation. Thus, the i in the Schrödinger equation is generator of rotations in a plane, and the normal to that plane is a spin direction. In other words, the Schrödinger equation is not describing a particle without spin; it is describes a particle in an eigenstate of spin, that is, with a fixed spin direction. Studying the implications of these facts has been a major theme of my research to this day. And more results will be published soon.
T: Great
H: Yeah, so, that keeps me going. .
T: And you’re still excited after forty years?
H: Yeah, that's right. So, if you are interested I tell you a little about what it has all lead to. Have you heard of zitterbewegung?
T: I’m not familiar.
H: That’s a German word meaning “trembling motion.” The term was coined by Schrödinger. He noticed that if you try to make a wave packet with the free particle solutions of the Dirac equation something funny happens. You can’t make a wave packet using only the positive energy solutions. The Dirac equation has troubles because there are both positive and negative energy solutions, and everybody believes that for a free particle the energy has to be positive. And, you need both positive and negative energy solutions to make wave packets, otherwise you don’t have a complete set. When you make a wave packet it has oscillations between positive and negative states that Schrödinger called zitterbewegung. The frequency of these oscillations is twice the de Broglie frequency. Do you know the de Broglie frequency?
T: Hmm!
H: It is mc squared over h-bar.. The zitterbewegung frequency is twice that, okay? Schrödinger
Thanks for the reference Hermann
Title: Geometric Algebra Explorer
Post by: hermann on August 22, 2017, 08:25:15 PM
I discovered the geometric algebra explorer:
https://gacomputing.info/ (https://gacomputing.info/)
Interviews can be found here:
https://gacomputing.info/blog/ (https://gacomputing.info/blog/)
Hermann
https://gacomputing.info/ (https://gacomputing.info/)
Interviews can be found here:
https://gacomputing.info/blog/ (https://gacomputing.info/blog/)
Hermann
Title: Zitterbewegung
Post by: hermann on August 30, 2017, 06:00:37 AM
David Hestenes famous paper on "Zitterbewegung" can be found here.
http://geocalc.clas.asu.edu/pdf/ZBWinQM15**.pdf (http://geocalc.clas.asu.edu/pdf/ZBWinQM15**.pdf)
Hermann
http://geocalc.clas.asu.edu/pdf/ZBWinQM15**.pdf (http://geocalc.clas.asu.edu/pdf/ZBWinQM15**.pdf)
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: jehovajah on August 30, 2017, 10:14:33 AM
http://youtu.be/qm5I_D9NN3g
http://m.youtube.com/watch?v=qm5I_D9NN3g
An abstract overview showing gradual application to real technological issues.
http://m.youtube.com/watch?v=qm5I_D9NN3g
An abstract overview showing gradual application to real technological issues.
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on August 30, 2017, 11:31:26 AM
Nice talk!
In it, this other one got especially highlighted:
https://youtu.be/LpqwdIoWgw0
In it, this other one got especially highlighted:
https://youtu.be/LpqwdIoWgw0
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on September 24, 2017, 01:55:36 PM
Hallo Kram,
thanks for the last video you have posted.
It reminds me on my Java Applets on color mixing and my Java Applet Fractal Generator which has brought me here into fractal forums.
Oracle and most of the internet browser no longer support Java-Pluggins for Java Applets.
So applications I spend a lot of work to implement them are can no longer be executed.
Which is very disappointing.
So it is not good when a technologie is controlled by only one company!
(http://www.wackerart.de/icons/mixer_icon_new.jpg) (http://www.wackerart.de/icons/color_palette_icon.jpg) (http://www.wackerart.de/FractalGraphics/Mandelbrot.jpg)
http://www.wackerart.de/mixer_english.html (http://www.wackerart.de/mixer_english.html) http://www.wackerart.de/rgb_palette.html (http://www.wackerart.de/rgb_palette.html) http://www.wackerart.de/fractal.html (http://www.wackerart.de/fractal.html)
I hope that one day Java Plugins will be again available for every browser.
Hermann
thanks for the last video you have posted.
It reminds me on my Java Applets on color mixing and my Java Applet Fractal Generator which has brought me here into fractal forums.
Oracle and most of the internet browser no longer support Java-Pluggins for Java Applets.
So applications I spend a lot of work to implement them are can no longer be executed.
Which is very disappointing.
So it is not good when a technologie is controlled by only one company!
(http://www.wackerart.de/icons/mixer_icon_new.jpg) (http://www.wackerart.de/icons/color_palette_icon.jpg) (http://www.wackerart.de/FractalGraphics/Mandelbrot.jpg)
http://www.wackerart.de/mixer_english.html (http://www.wackerart.de/mixer_english.html) http://www.wackerart.de/rgb_palette.html (http://www.wackerart.de/rgb_palette.html) http://www.wackerart.de/fractal.html (http://www.wackerart.de/fractal.html)
I hope that one day Java Plugins will be again available for every browser.
Hermann
Title: Re: Geometric Algebra, Geometric Calculus
Post by: kram1032 on September 24, 2017, 02:55:53 PM
Try migrating/porting to Javascript and HTML5 over Java and Flash. That's currently the future of the web. You can also try using fancy WebGL for graphics stuff to speed things up a lot.
This thread has been very long running and while it has slowed down it appears to still generate interest. I was wondering what we should do about it for the new forum.
Bringing over all 15 pages seems a bit much. And even if we cut out the fat, it'll still be a lot. Probably just retire this thread and open up a continuation thread in the new forum?
This thread has been very long running and while it has slowed down it appears to still generate interest. I was wondering what we should do about it for the new forum.
Bringing over all 15 pages seems a bit much. And even if we cut out the fat, it'll still be a lot. Probably just retire this thread and open up a continuation thread in the new forum?
Title: Re: Geometric Algebra, Geometric Calculus
Post by: hermann on September 24, 2017, 05:50:11 PM
Hallo Kram,
my Java Applets contain many weeks of work and ii is demotivating, that this work is lost.
My fractal gallerie was a bit unic it does not display the fractals as JPG pictures but obened a Java window from which it was possible to explore the selected fractal further.
For this I used already JavaScript technologie. I am working at the moment on javascript technologie for visualisation.
The tables in my prime number pages http://www.wackerart.de/mathematik/primzahlen.html (http://www.wackerart.de/mathematik/primzahlen.html) are generated by JavaScript.
I also think about using a JavaScript 3D Library for visualisation.
But this will be new material. I hate reprogramming functionaly for which I already have a good implementation.
I still have a deep interest in geometric algebra. So we should continue geometric algebra in the new forum.
We also should find a good and compact introduction to show the basic freature of geometric algebra.
Hermann
my Java Applets contain many weeks of work and ii is demotivating, that this work is lost.
My fractal gallerie was a bit unic it does not display the fractals as JPG pictures but obened a Java window from which it was possible to explore the selected fractal further.
For this I used already JavaScript technologie. I am working at the moment on javascript technologie for visualisation.
The tables in my prime number pages http://www.wackerart.de/mathematik/primzahlen.html (http://www.wackerart.de/mathematik/primzahlen.html) are generated by JavaScript.
I also think about using a JavaScript 3D Library for visualisation.
But this will be new material. I hate reprogramming functionaly for which I already have a good implementation.
I still have a deep interest in geometric algebra. So we should continue geometric algebra in the new forum.
We also should find a good and compact introduction to show the basic freature of geometric algebra.
Hermann