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Author Topic: Polynomial rotations  (Read 18476 times)
Description: Unary operators in polynomial form represent geometric space
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« Reply #30 on: August 20, 2011, 02:26:41 PM »

In the study of space i have to distinguish 2 iterative and complementary processes. The first is the action of motion on regions however small the scale. The second is the post processing of the resultants of the action by my computational perceptual output processes.

This duality exists fundamentally throughout the whole of my experiential continuum. There is a reality i cannot know except by a model which i experience as knowing. Thus the action of motion on regions  in space in their exactitiude are not knowable, but an approximate rendering of surfaces and perspectives is possible by taking a pragmatic approach to the actual exact data. Thus a processing schemen may use latest data point, average datapoint, median data point, statistical average data point, moving average data point, the variance etc in premising its processing model, thus determining its output.

These methods act as filters to raw data which is already heavily filtered by the data collection process. thus i arrive at a statisitical description of the raw data , thus requiring my model to be a statistical model, and my appreciation of the experiential continuum to be  a statistical process producing a statistical output from statistical data. My knowledge of reality is therefore purely statisitical, deriving from iterative processes within me reacting to iteraive processes of motion without me acting on regions of space in a way i can only sample statistically.

Things therefore are never quite what they seem. In this sense spaciometry or geometry is my best "average" of the whole scheme of things.

the precision i can describe here with mathematical language, has to be rendered in perceptual language to give the actual experience. The fractal generator enables this to happen in the most remarkable way, and consequently gives real insight into the spaciometry of my reality. For those not familiar with vectors or tensors, the fractal generator provides a hands on , sandpit way to grasp the essentials of these notions, for what it is worth. However even a simple geometrical knowledge will produce the most exquisite results.

Those who have worked on the formulae and those who have worked on the application have produced our most fundamental representative model of space as we experience it.

motions in space at all scales with surface computations at all scales is about the nub of it. A complex vector algebra, quaternion algebra, clifford algeba and Grassmann algebra are all the different ways the app can be used to describe relations of motion sequents in space. However, we can see, hear or feel nothing without the surface processing algorithms that output surfaces, the auditory processing algorithms that output sound intensity,and the haptic processing algorithms that output force intensity.

The possibilities are truly infinite.
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« Reply #31 on: November 12, 2011, 11:39:51 AM »

I have long entertained the notion of a rotation vector, but that was because i did not know the origins of the word vector coined by Hamilton, and pinched by all and sundry but particularly Gibbs, who used it to deliver a glancing blow to Quaternions! Oh the gentlemanly misconduct of Hamilton's times! wink
In any case, my increasing understanding leads me toward the notion of "path" or "arc". Very traditional terms i know, but many times the old is better! cheesy

I can define a "rotating vector". but the notion of a rotation vector does not seem to fit that tool. And so i recognise the path of the tip of such a rotating vector is what i am after. Similarly, tangential vectors may envelop a path their envelope tracing out the path.  The path around the unit circle is of particular significance: i can use it to measure rotation in the plane. Thus if U is the radian arc of a unit circle, i can define a general path c*U which takes me around the circle c radians. I rotate c radians around the unit circle path. However, if i leave the unit circle , the path P of the radian arc becomes a length in whatever units the radius is measured, c*P still takes me round the larger circle, and c now scales an "angle" ratio, but the Arc length has to be remembered to calculate circular distance traveled in units, not radians. This is made simple by the system, because P is always = r the radius of the circle in question so that the ratio is always reducible to 1:1. Thus the scaled angle ratio * radius gives us the distance around the circumference of a circle in the units the radius is measured in.

I can then define a path P for a circle as being an arc equal in length to the radius of a circle, and a general circle path length  ΩP being of magnitude Ω radians *r the radius of the circle in its units. Thus a Path for a circle is the same as a radian only in the unit circle, but of course is in a ratio to radians of the unit circle.

Now i can add a path to a vector v so that v+ΩP= w where the magnitude of |w|= |v|

Clearly there is a vector x such that v+x = e*w=W where e is some scalar, and i can now divide |W| by |w| to give e, in a straightforward  fashion.
|W/w|=e= |(v+x)/(v+ΩP|)
giving in this case
|w|*|(v+x)|= |W|*|(v+ΩP)|: we can consider vector magnitude products as a vector magitude producted with a common vector and a circular path. There is no sense in going any further, as this is not a value enumeration, but a note to include the rotation effect in producting vectors.

One other thing to note is that the scalar e scales the vector v and the path ΩP, as one would expect. The effect is to change P to e*P leaving Ω the same.

Producting is derived from combinatorics, and is the consideration of how to relate and place one form against or with another. Thus the * and the plus operations need to be defined before assuming any meaning. I have not done so here to illustrate that "operations" often cause confusion because we think addition and multiplication are number bonds! They are not number bonds they are "combinatorial bonds" and define combinations.

The insights gained from the above manipulation will need to be expressed rhetorically until a good notation is devised to express it symbolically.
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« Reply #32 on: November 12, 2011, 03:46:48 PM »

I learned a lot through the difficulties i explored in this thread, especially the difference between the rotation and the axis, but i did not understand vectors or modern vector calculus when i started. That is not a bad thing, because i think maths is poorly taught, as a set of skills not as a philosophical life style. Mathematicians today are just that, mathematicians, whereas in all of the early history of maodern maths mathematics was done by people who were more rounded : Philosophers, classical scholars, Administrators and Educators with a broad curriculum view. Mathematics, i know has lost its way, chiefly by being stripped of its true nature that is natural philosophy within the context of theosophy and theurgical practice.

Hamilton, when he formed the notion of vector, and then fell upon the idea of quaternions did so in the hopes of serving his god's great purpose. It mau be remarked that he was not unique in doing this, as the same trait is found in Grassmann, Euclid , Pythagoras, Gauss and of course Newton, to whom the laws of gravity were given "as if he were god's messenger, god's angel to mankind."

Whatever the inspiration, i have studied sufficiently to know that mathematics has been stripped of its essential nature, and its purveyors are feeding us drivel and stones instead of milk and bread and yes honey too.

http://docs.google.com/viewer?a=v&q=cache:x4zsghc8SoIJ:www.lrcphysics.com/storage/documents/Hamilton%2520Rodrigues%2520and%2520Quaternion%2520Scandle.pdf+quote+james+Maxwell+on+quaternions&hl=en&pid=bl&srcid=ADGEESjlbAVlGbzNNu8exy3jO1io5Py5XQwyZXZacV0WSkmPIVeaHzVZWG34Q17QvasPYTUs1r6MpMM6ZmX3549Gjb1c-oaNPAECpNNpNXes8GmFFrBiPzgCHNpZJ0mhAW9MApQgPwcl&sig=AHIEtbRnj3jj4Ddqjc5EX2axH6KuuB3RCA.

I hope you enjoy this link, which sets out what i had hoped to do in this thread in its hitorical context, and shows the remarkable and clear insight that our fathers had, which has been denied us in our mathematical education.

I lament, a bitter lamentation, that when i asked why fractions and ratios in our school text books that no one could tell me the fundamental nature of just these 2 things in the development of the vector algebras of today, nor the absolute and generative insight that Newton derived from just these 2 in grounding his calculus of fluxions, the differential calculus. The riches of just these 2 things go on and on and into a lifestyle of just proportionality and requisite fairness as advanced by Pyhagoras.

It is therefore with the greatest admiration that Benoit Mandelbrot can be seen to have reversed the trend by concentrating on these 2 fractions, from which he derived fractal, and ratios from which he derived "almost self similar". His work naturally sits amongs a great field of vectors and other algebras and provided entrance for computers into the very stuffy heart of mathematicians, reinvigorating the beauty and dropping from their eyes the scales by whiich they declared beauty "monstrous", repetiton and iteration overwhelmingly "boring", and instant public communication "dumbing down."

Polynomial rotations may be a backwater issue in the grand scheme of things, but i am glad i attempted to do it, inspired by the great energy exhibited on the forum for ways to understand space so we could create the Mamdelbulb. The insights here have helped me to grasp the more difficult presentations in unfamiliar notation elsewhere, and also revealed where bombast has taken over from common sense!

The combinatorics here are but one of many possible combinatoric models, but no combinatoric can be of any sensible use without the form and the sequences to which it refers. The forms are simple, the sequences natural, the combinations understandable, and the many iterations do-able. They do not need to be obscured in gibberish.
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« Reply #33 on: January 01, 2012, 12:07:09 PM »

The triples or triplets which i sought to explore in this thread, having a burning question as to why had it not been fully explored, but specifically would it hold the searched for holy grail of the Mandelbulb, i have at last come upon a lucid account of Hamilton's researches into the very same.

Usually and annoyingly these points are presented as if incontestable fact, and alien to human sentiment. Yet the human scientific endeavour, hope brimmed , is much more informative and affective, revealing at once the doubts and uncertainties and triumphs that necessarily precede development of ideas. And indeed Hamilton was never satisfied with the revelation so far of his topic called quaternion.

http://encyclopedia.jrank.org/PYR_RAY/QUATERNIONS.html Follow the link and enjoy.

My own ideas here are related but different and are a treatment of the subject in analogy. Although some results may attain the general treatment is far more restricted than Hamilton's or Grassmanns and illustrates the assumptions that have to be made , in the light of having a fractal generator app to test results, in order to achieve calculable and manageable actions in space.

What i learned here was the difficulty that notation presents, and the falsity of many taught assumptions and expectations, but the joy of being able to create notation that actually worked in part.

Now, because i have a better apprehension of a tensor/vector algebra not as some abstract principles, but as everyday apprehensions and interactons of and with space, i recognise this as an attempt to form a complex vector algebra in assumed continuous space.I also recognise all these expressions as arising out of notation to describe relationships within a form in space, and as such a direct and immediate descendant of Euclid's Stoikeioon. I make the bold claim that in fact Euclid both in his Stoikeioon and in his other Books had substantially worked out the laws of spatial relationships and actions, and that Appolonius to Archimedes , with Ptolemy and Eudoxus sees a refinement fit to purpose.

What we have been forced to do is to reinvent the wheel! However, that is not a bad thing. What is bad is to think higher of ourselves than we ought, because it is evidently demonstrated that our forefathers "knew a thing or two"!
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« Reply #34 on: August 19, 2012, 10:30:11 AM »

If i think of a Quaternion as a space attached to an external origin, that is a reference frame for a space with a position vector for the reference frame in some external reference frame, then the quaternion is a natural candidate for relative comparisons.

Now this is not "relativity" in the sense of the relative speeds. this is relativity in terms of frames of reference, each quantity in the quaternion being a "vector " in an external frame. This gives a complete local reference frame description in an external reference frame. As an example. if i am the origin of the external reference frame, then i give a loal reference frame to every object in my reference frame. Keeping it simple: when i observe a box of tissues i can immediately give the box its edges as its local reference frame, only taking the edge vectors from one corner, and distinguishing that corner by a position vector from me to it.

Now of course the constraints on the box have to be specified, and so i immediately get a lagrangian description of the relative situation. This kind of Lagrangian is of course called a Hamiltonian.

Clearly everything is specified except the orientation of the box in my reference frame, and to do that i need at least one more position vector from me to another corner of the box. However, if i do not do that then the box is free to rotate in my refernce frame.

Suppose now i have two such quaternions , two boxes in my space, what does quaternion addition mean? What does quarernion "multiplication " mean?

The underlying e;ements of this quaternion description are vectors, not so called scalar quamtities, so addition in this case means vector "addition". and so multiplication  means what?

This nicely illustrates the point that addition and multiplication are the "wrong" notions we get taught from early childhood. We only ever are talking about synthesis and antisynthesis, combinations of forms with regard to their relative orientations , positioning and quality of fit. So i do not want to know about multiplication, i want to know about factorisation: wha components fit where to make the whole and what are there relative positions. Thus the factor forms tha come out of any "multiplication" actually are an invitation to work ou what forms they could possibly be part of.

We make life easy on ourselves generally by chosing a form with high symmetry like a cube or a square or a sphere, and then we do ot need to worry about the form, and generally we do not need to worry about the orientation, because of symmetry, and so we use a cube or a square as a standard form.

However, in this case we cannot ignore these aspects of the details of a combinatorial expansion. for they do not necesarily refer to a cube, or if they do the cube is not necessarily uniform, in the sense that a cubes faces are all pointing in the correct orientation!

Doug Sweetster gives the current notation for quaternion product which involves the cross and dot products in its expansion
 http://www.theworld.com/~sweetser/quaternions/intro/multiplying/multiplying.html

When i interpret this way of notating a quaternion i see straight away that it produces a magnitude in my space, lets say a new position vector, three vectors , lets say for the vectors of the edges of the new volume and a cross product which signifies a rotation axisfor elements inthe new volume. The size of the volume can be determined by the wedge product of the edge evectors

So in general the product of thes two boxes in my reference frame is a box in my reference frame that has a volume, edges and an axis of rotation which is a local axis for its elements, and a new position vector in my reference frame.

This position vector is derived from the 2 position vectors of the generating quaternions and for simpliity we denote that direction as e hat, so the position always is in e hat, but what if it were not?

Well i havenot got to that yet! However, keeping e hat the same makes this version of quaternion product ideal for spacetime descriptions also. However, you will note "time" is dependent in some complex way on space variables.
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« Reply #35 on: August 24, 2012, 01:14:34 PM »

I did say I was pleased I tackled this topic, and now I have found some clarity in Hamilton's own work. The problem I had was the confusion between orientation and the unwary operator idea I started with. As it turns out the unwary operator idea is a way of trying to understand some simple combinatorial construction rules which few understand, despite running through them as properties of groups and rings etc.
Suffice it to say that Hamilton was not unclear about what were operators and what they were operating on, and I was and always has been a Hamiltonian vector in the quaternion group structure, and also in the couples group or field structure as Hamilton defined it. Thus the famous i2 =-1 was a statement about a hidden operation which was rotation. The confusion in calling it multiplication arises from its past use and a less than careful distinction between multiplication and algebraic multipling. A distinction Hamilton took care to make.
Algebraic multiplying in couples was making effective steps , but in quaternions it is effective rotations. The effective steps in the plane involved a step a rotation through pi/2 and then another effective step in the new direction. In 3d using quaternions the effective step is a step then a rotstion through pi.

Due to De Moivre and the Cotes Euler identity any direction can be moved in, any orientation accessed by the quaternions.
Dropping back to couples, it is commutativity which makes them appear different to quaternions, but in fact whatever operation applies to quaternions also applies to couples, that is complex numbers. So any rotstion is effected by a pre and post multiplication operation, when this is done , rotstion of the couples is also through pi. In general, pre and post " multipling"  doubles the angle of rotation, so half angles should be used.
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« Reply #36 on: August 24, 2012, 09:31:54 PM »

i0j0v0:i10j10v10:i20j20v20:i30j30v30
i0i00i00i10:i10j10v10:i20i20i30:i30j30v30
j0j00j00v00:i10j10j10:j20j20v20:i30j30j30
v0i00j00v00:v00v10v10:i20j20v20:v20v30v30
----------------
ii10i10i20:i20XX:i30i30i00:i00XX
jj10j10X:Xj20j20:j30j30X:Xj00j00
vXXv10:v10v20v20:XXv30:v30v00v00
----------------
i2i20i20i30:i30XX:i00i00i10:i10XX
j2j20j20X:Xj30j30:j00j00X:Xj10j10
v2XXv20:v20v30v30:XXv00:v00v10v10
----------------
i3i30i30i00:i00XX:i10i10i20:i20XX
j3j30j30X:Xj00j00:j10j10X:Xj20j20
v3XXv30:v30v00v00:XXv10:v10v20v20

A table of the resultants of associative actions of unary operators on orientations

i00=i0
j00=j0=i00
v00=v0=i10


I have adopted a convention that a rotation  produces its resultant orientation when that lies in its plane of operation, but an association of rotations is replaced  by its rotational equivalent in the plane.

under construction


This was my best shot, and I was getting closer to discovering Hamilton's triples, work he did before finally realising he needed a fourth calculation axis.

At this point I had nearly everything in place, I was just confused by my use of the term associativity. This was based on but not the same as the group associativity axiom, the combinatorial rules for group elements.
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« Reply #37 on: August 25, 2012, 12:36:34 AM »

In my blog I carefully outline the rotation actions in the quaternions that I identified here as I,j,v. I clearly need to correct some of the tables in light of this.

http://jehovajah.wordpress.com/jehovajah/blog/quaternion-8-clarifying-the-rotation-action

This also helps in programming the results for Quasz.
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« Reply #38 on: August 29, 2012, 07:59:52 AM »

At last! Hamiltons bubbles burst in iridescent hues in my feeble brain!
http://jehovajah.wordpress.com/jehovajah/blog/2012/08/28/hamiltons-bubbles
Polynomial rotations have got it wrong but so nearly right! embarrass
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« Reply #39 on: April 28, 2013, 09:50:51 AM »

The conception of polynomial rotations was born out of an idea of unitary operators. I have long since abandoned that idea as I studied Hamilton and then later Grassmann.
Despite my protestations I have been dragged unwillingly into the ring and group theory group of mathematicians, the so called abstract algebraists.

Well as you can imagine I am not going to drown in obscure languages and jargon. I am not going to change a thing in this thread, because it serves to show how heuristics works, and how bashing your brains out with algebra which literally means in Arabic "the twisting" referring to the Indian method of reconciling things like -1 *-1=+1, which is not now they considered it at all! But it gets the point across I hope

Well this brain twisting actually does some good if you stick with it. I can actually trace back to this thread some ideas I have in regard to fluid dynamics and vortices!
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« Reply #40 on: April 29, 2013, 12:35:47 AM »

Hi,

I worked too on the same subject as Tim Golden and Kujonai, and I have some threads with Tim.
I have proposed the absolien numbers last year on that site, wich are 廦uivalent to polysign numbers (as I interstood later), but with matrix formalism more simple I think : vectors with only positive values, it can be seen as generalization of signs, or the end of signs !

I believe that absoliens are isomorph with quotien ring algebras, and so are very related to polynoms and multicomplexes MCn.
I am interested with your opinion. My site is:     https://sites.google.com/site/yannispicart/   (See page absolien numbers).
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« Reply #41 on: April 30, 2013, 10:45:34 PM »

Hi Yannis.
Tim and Kujonai what a blast.
I have looked over your site  and tried to follow your presentation.
The detail is a bit mind blowing in full mathematical garb, but it is essentially what I expected. The Matrix notation I mentioned to Kujonai in passing, in the form of a question, but I was too bogged down to pursue his work in depth.
I did a bit on the Quaternion 8  group and identified where Hamilton had made a "mistake" iGmetryeoms of brute force making Quaternions work, but apart from a better understanding of reference frames and solving the Hamilton triples I have not done a lot since.

I am mulling over the Vortex 9 and vortex 18 groups very slowly, but I have no reference frame to represent them..

Kujonai gave a Kujonai 27 group which I had a go at, but it too lacks proper reference frame to represent it.

The prior art to all of this is found in the work of Sir Roger Cotes and Augustus De Moivre students of the great Newton. The notion of the roots of unity are absolutely the bedrock of these types of attempts, including Musean numbers etc. for this reason I abandoned the interesting poly sign work and concentrated on the roots of unity.

However my puzzle really was to grasp why sqrt(-1) was so mysterious. It turned out for me that the whole concept of numbers was misleading, and the whole foundation of mathematics was bent out of shape.

The absoliens the roots of unity the Quaternions etc all turned out to be special cases of a more general group or ring algebra. The way of working to this conclusion, in modern times was pioneered by the Grassmanns,

This does not detract from your work. The detail you have explored is the necessary research to build this model. .

The thing that interests me is of course removing the contra sign. This was always possible but it is a pyrrhic victory when the sign is replaced by a more complex symbol that does the same job.

Contra and subtraction are similar but contra is a directional status while subtraction is a dismembering process.

The homemorphism to the cyclic groups is also well studied.

The trick is to build a tool to display n dimensions. Using simplexes for 4 dimensions  is also an explored topic..

I am going to take my time looking at your work to help me with the V9 group.

The anti ontology is a topic most do not consider. Basically it means if A exist then notA also exists, in a universal set diagram that means the complement of a exists if A exists.. I have also called this conjugation.

While I think a quotient ring for Quaternions exists, that is a division algebra, I also think commutative Quaternions exist also in a division ring.. The non commutative Quaternions arise because Hamilton did not recognise the conjugate of k in his famous rules!
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« Reply #42 on: May 23, 2013, 10:10:24 AM »

The thread here is a fundamental assessment of the geometric algebra that underpins the notion of rotation.
http://www.fractalforums.com/new-theories-and-research/geometric-algebra-geometric-calculus/msg61550/#msg61550

There is a lot more to come.

The restrictions i put on these Twistors in the early days were misguided, and the reason why i could not then achieve the vision i had. Since then i have solved the issue through Newtonian Triples, which is a model built from a much broader Method Developed by Hermann Grassmann.

Funny how things turn out! cheesy
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« Reply #43 on: September 12, 2013, 12:18:15 PM »

Yes, there is life in this old topic yet!

Just updated a few posts in the past with some clarifying remarks and highlighted Norman Wildbergers treatment of Euler Rotations. Pay attention to the doubling effect.
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May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
jehovajah
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« Reply #44 on: August 23, 2017, 03:21:06 AM »

Just found an application for some of the ideas in this thread on my thread in Magneticuniverse.com !
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May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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