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 Pages: 1 2 3    Go Down       Author Topic: The "operator" i is more complex than that!  (Read 13493 times) Description: numbers and things as roots of unity and complex factors 0 Members and 1 Guest are viewing this topic.
jehovajah
Global Moderator
Fractal Senior      Posts: 2749 May a trochoid in the void bring you peace  « Reply #45 on: July 15, 2017, 05:35:29 PM »

If I consider the Quaternion form as 3 circular quarter arcs , I need to specify that adding arcs is done by ensuring the diameters are parallel and the arcs add as in vectors head to toe , or the end of one to the beginning of the next.
This in fact produces a step like firm that moves a point diagonally in a plane .
To obtain a circular movement we have to specify that the second quarter arc has its diameter rotated orthogonally to the first before continuing. This is properly and analogously represented by a composition we may consider as multiplication .
Now in addition we have to define the composition of quarter turns which though continuous / contiguous are not in the same plane. .
So I + I is 2I but I x I is I2 a quarter turn of the diameter in the plane of the circle .
If J is a quarter circle arc in an orthogonal plane then J x I or I x J requires a quarter turn of the diameter into the second plane.
We need to define the direction of this quarter turn and the order in which the process is carried out.
These quarter turns of the diameter are not notated and this has been the problem historically. By not defining the composition rigorously and clearly no one understood, and indeed were mystified by the concept of multiplication! To be sure multiplication is a Designed process, it always has been, but we have been taught that it was a natural gd given function of numbers! Nothing could be farther from the truth!

Bearing ths in mind we can design a Quaternion using 3 orthogonal diameters from which quarter arc vectors are described and a lineal vector to the point of intersection of these 3 diameters. That lineal vector may be used to determine the radius of the quarter arc vectors as well as to provide a rotational Axis to drop peroendiculars onto,
This way we connect a Quaternion to the fundamental lineal orthogonal axes we have accepted as standard.
We o not have to accept that as a standard, but we must agree some conventional standard. .

When I started the thread polynomial rotations I struggled to grasp ths simple process definition, and if course the confusing notation
Now here's.the rub: we do not have to use quarter turns! We can use any fraction of a circular perimeter as a circular arc vector and the planes in which they operate do not need to be orthogonal! .

The Cotes DeMoivre theory of the roots if unity deal with this concept using polynomial depictions , but what I am describing is a straightforward spaciometric construction.

Whether we confine the circular vectors to a sphere or a composition of planar circular arcs we can describe motions of points by lineal or circular arc vectors ?
In addition if I is the arc vector for a unit diameter  2I is the sum of 2 of them but will give a resultant point equivalent to R if that is the  arc vector for a diameter of twice the unit diameter. Logged

May a trochoid of ¥h¶h 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
Global Moderator
Fractal Senior      Posts: 2749 May a trochoid in the void bring you peace  « Reply #46 on: July 17, 2017, 04:15:43 AM »

The addition and multiplication of circular arc vectors generates trochoidal motions or positions, the addition rule can be modified to allow the radial phase vector to be attached to the radial vector of the next circular arc tip to toe. Thus the position or motions trace out the familiar trochoidal patterns.
The usual exponential form of a circular arc vector is based on this type of circular arc addition. Thus the Fourier series in exponential form is a trochoidal line pattern .
Using three Fourier series can capture a 3d trochoidal description of space.
However a Quaternion Fourier series or a Grassmann Twstir depicts a trochoidal surface pattern based on interacting spherical surfaced.

To depict saciometrically a volume dynamic a Quaternion Fourier transform would be naturally descriptive, but the mathematical behaviour of Grassmann twistors is poorly understood and poorly researched.

There is one application of a Quaternion Fourier transform and that is in image compression . So far this work has been pioneered by just one person in the main. Logged

May a trochoid of ¥h¶h 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
Global Moderator
Fractal Senior      Posts: 2749 May a trochoid in the void bring you peace  « Reply #47 on: July 17, 2017, 06:30:05 AM »

This is a quote from an interview with David Hestenes in 2009

Quote
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

What is important is that i is identified as having a geometric interpretation. Hestenes does not refer to the work of the Grassmans directly but we know that Hermann Grassmann based on Eulers work had already made this identification in the Vorrede of the 1844 Ausdehnungslehre . And of course his use of " vectors" inspired Caleys Matrix representation of many of his designed and defined products.

Now I have identified i as the quarter circle arc, a circle defines a plane, so a circular arc defines a plane of rotation .paulis sigma algebra defines a sphere of rotation as a unit volume which spins in the i pland of rotation
Quantum physics is about trochoidally dynamic surfaces . It differs from so called classical Einsteinian physics as a straight line differs from a curve! On the very large scale and removing unwarranted restrictions like absolute speed or velocity the two descriptions can be made to agree.
On the very small scale a straight line can not approximate a small curve ! Or rather a large curvature. On such a scale classical physics becomes too complex and too slow in computation .
When Newton tried to define a fluid dynamic version of space, he found the computation too onerous. Simplifying assumptions he made and the fundamental assumption that fluids only act resistively led to innacuracies that just were not there in the point mass model.

The point mass model was based on one keen observation. Motion tended to create a point or centre of symmetry. By reducing mass to a point he could ignore all the extraneous motions in his quantity of motion and identify the position of a body. What he could not do is explain the real world fluid dynamic motion of such a body because much of that motion is rotationl and vibrational and exhibits as different forms of energy to put it simply.  Thus in his fluid dynamics he failed to account for counter rotations within and around defined bodies, and so there impact on the resultant motion. Le Sage provided an important analysis of the effect of some environmental I pacts, but he did not appreciate the contribution of rotationl dynamics in fluid dynamics, nor the presence of magnetic behaviours in space as we now know.

All these issues can be addressed with modern computational means and new mathematical developments of forms Logged

May a trochoid of ¥h¶h 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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