Logo by blob - Contribute your own Logo!

END OF AN ERA, FRACTALFORUMS.COM IS CONTINUED ON FRACTALFORUMS.ORG

it was a great time but no longer maintainable by c.Kleinhuis contact him for any data retrieval,
thanks and see you perhaps in 10 years again

this forum will stay online for reference
News: Follow us on Twitter
 
*
Welcome, Guest. Please login or register. March 28, 2024, 02:25:03 PM


Login with username, password and session length


The All New FractalForums is now in Public Beta Testing! Visit FractalForums.org and check it out!


Pages: 1 ... 12 13 [14] 15   Go Down
  Print  
Share this topic on DiggShare this topic on FacebookShare this topic on GoogleShare this topic on RedditShare this topic on StumbleUponShare this topic on Twitter
Author Topic: Geometric Algebra, Geometric Calculus  (Read 11716 times)
0 Members and 1 Guest are viewing this topic.
macawscopes
Forums Freshman
**
Posts: 15


« Reply #195 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
Logged
hermann
Iterator
*
Posts: 181



WWW
« Reply #196 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
Logged

macawscopes
Forums Freshman
**
Posts: 15


« Reply #197 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.  x^2 + xy - y^2 is a homogenous polynomial of degree 2.  Here's a homogeneous polynomial of degree 4: x^4 - x^2y^2 + y^3x.

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.  x^3 - y^2z + xyz) 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!
« Last Edit: April 17, 2017, 06:48:40 AM by macawscopes » Logged
hermann
Iterator
*
Posts: 181



WWW
« Reply #198 on: April 17, 2017, 07:11:50 PM »

But it is possible to build the geometric product of three vectors, so what is new?
Logged

macawscopes
Forums Freshman
**
Posts: 15


« Reply #199 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.




Logged
hermann
Iterator
*
Posts: 181



WWW
« Reply #200 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:

<a href="https://www.youtube.com/v/Tw8w4YPp4zM&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/Tw8w4YPp4zM&rel=1&fs=1&hd=1</a>

I also found some information on Spinors and Trialities on the hompage of John Baez.
http://math.ucr.edu/home/baez/octonions/node7.html
May be this helps a bit.

Hermann
Logged

jehovajah
Global Moderator
Fractal Senior
******
Posts: 2749


May a trochoid in the void bring you peace


WWW
« Reply #201 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.
« Last Edit: July 17, 2017, 04:46:37 AM by jehovajah » 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!
hermann
Iterator
*
Posts: 181



WWW
« Reply #202 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
« Last Edit: May 01, 2017, 02:53:54 PM by hermann » Logged

hermann
Iterator
*
Posts: 181



WWW
« Reply #203 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
Very intresting are his talks on Zitterbewegung and his Interpretation of Schrödinger and Dirac equations.

Hermann
Logged

hermann
Iterator
*
Posts: 181



WWW
« Reply #204 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
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
« Last Edit: May 22, 2017, 02:54:51 PM by hermann » Logged

hermann
Iterator
*
Posts: 181



WWW
« Reply #205 on: July 08, 2017, 03:42:32 PM »

The following lesson from Norman Wildberger may be helpfull for Macawscopes Question on trible products:

<a href="https://www.youtube.com/v/e17J0mOhelQ&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/e17J0mOhelQ&rel=1&fs=1&hd=1</a>

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

Hermann
« Last Edit: July 08, 2017, 03:50:09 PM by hermann » Logged

jehovajah
Global Moderator
Fractal Senior
******
Posts: 2749


May a trochoid in the void bring you peace


WWW
« Reply #206 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
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!
hermann
Iterator
*
Posts: 181



WWW
« Reply #207 on: August 22, 2017, 08:25:15 PM »

I discovered the geometric algebra explorer:
https://gacomputing.info/

Interviews can be found here:
https://gacomputing.info/blog/

Hermann
Logged

hermann
Iterator
*
Posts: 181



WWW
« Reply #208 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

Hermann
Logged

jehovajah
Global Moderator
Fractal Senior
******
Posts: 2749


May a trochoid in the void bring you peace


WWW
« Reply #209 on: August 30, 2017, 10:14:33 AM »

<a href="http://www.youtube.com/v/qm5I&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/qm5I&rel=1&fs=1&hd=1</a>

http://m.youtube.com/watch?v=qm5I_D9NN3g
An abstract overview  showing gradual application to real technological issues.
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!
Pages: 1 ... 12 13 [14] 15   Go Down
  Print  
 
Jump to:  

Related Topics
Subject Started by Replies Views Last post
geometric spirals Mandelbulb3D Gallery bib 0 836 Last post October 25, 2010, 09:48:32 PM
by bib
Geometric Buddha Images Showcase (Rate My Fractal) John Smith 0 983 Last post June 07, 2012, 09:05:17 PM
by John Smith
Retro Geometric Still Frame FracZky 0 1127 Last post May 02, 2013, 07:30:42 PM
by FracZky
Geometric Patterns No. 2 Saturn&Titan Gallery element90 2 816 Last post February 09, 2014, 03:43:35 PM
by Dinkydau
Geometric Fractals Help Help & Support lancelot 13 795 Last post December 15, 2014, 06:10:29 PM
by Sockratease

Powered by MySQL Powered by PHP Powered by SMF 1.1.21 | SMF © 2015, Simple Machines

Valid XHTML 1.0! Valid CSS! Dilber MC Theme by HarzeM
Page created in 0.733 seconds with 24 queries. (Pretty URLs adds 0.039s, 2q)