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Author Topic: Geometric Algebra, Geometric Calculus  (Read 11762 times)
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kram1032
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« 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 w+\left(x|y|z\right) 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):
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.

(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)
« Last Edit: August 16, 2014, 06:32:16 PM by kram1032 » Logged
kram1032
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« Reply #1 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.
« Last Edit: May 15, 2013, 06:32:10 PM by kram1032 » Logged
jehovajah
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« Reply #2 on: May 17, 2013, 09:42:29 AM »

Do not be so fatalistic Kram!
My Newtonian triples are based on these type of notations!
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kram1032
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« Reply #3 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
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« Reply #4 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.

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.


« Last Edit: May 23, 2013, 03:17:33 AM by jehovajah » Logged

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kram1032
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« Reply #5 on: May 22, 2013, 02:50:39 PM »

That's not quite what I had in mind but thank you very much. smiley
I also didn't think you'd jump at this so quickly...
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« Reply #6 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/

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?
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kram1032
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« Reply #7 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)

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:


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:


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)
« Last Edit: June 01, 2013, 11:34:40 PM by kram1032 » Logged
kram1032
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« Reply #8 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 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)
« Last Edit: May 23, 2013, 01:57:29 AM by kram1032 » Logged
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« Reply #9 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
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« Reply #10 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. 
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kram1032
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« Reply #11 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.
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« Reply #12 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:



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.
« Last Edit: April 13, 2014, 10:30:59 PM by kram1032 » Logged
jehovajah
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« Reply #13 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!
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kram1032
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« Reply #14 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...
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