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Geometric Algebra Discussion Group



Geometric Algebra Discussion Group - LiveJournal.com



Last Build Date: Thu, 11 Jun 2009 20:42:58 GMT

 



Orthogonal Transformations with Complex Numbers, Quaternions, and Clifford algebras

Thu, 11 Jun 2009 20:42:58 GMT

Cross-posted from (image) mathematics

A good friend of mine recently discovered some of the fun things you can do with complex numbers if you're using them to represent points in the plane. Shortly thereafter, I re-read a passage by Tony Smith about why one should be interested in Clifford algebras. Tony Smith's passage included all of the fun one can have with the complex plane and extends it to three, four, five, and more dimensions. I thought, I should segue from the complex numbers in the plane to Clifford algebras to quaternions in 3-space to Clifford algebras again in a series of posts on my website.

I have posted four articles in that series so far:




Plane Transformations with Complex Numbers and Clifford Algebras

Mon, 08 Jun 2009 17:09:05 GMT

cross-posted to (image) mathematics

A good friend of mine recently discovered some of the fun things you can do with complex numbers if you're using them to represent points in the plane. Friday, I re-read a passage by Tony Smith about why one should be interested in Clifford algebras. Tony Smith's passage included all of the fun one can have with the complex plane and extends it to three, four, five, and more dimensions. I thought, I should segue from the complex numbers in the plane to Clifford algebras to quaternions in 3-space to Clifford algebras again in a series of posts on my website.

I have posted two articles in that series so far:




Dead community!

Sat, 11 Mar 2006 09:46:36 GMT

This place is so inactive! Nobody ever comes here anymore ?




Wed, 18 Jan 2006 09:24:42 GMT

Firstly I'm new to this community, but I love Algebraic Geomtery immensly as you can see from my userpic/logo, Grothendieck's Motives letter to Serre. (That should be familiar to most of you) Anyway I was wondering if we could add a chat room to the community for live discussions, like yahoo perhaps that way we can discuss the answers for questions posted here and resulting answers with refer to it. Would help understand the problem more clearly.
I'm starting off with my first problem set:

Let x_1,...,x_n be points in R^2, and let f:R^2-->R be a function such that for any rigid motion T in the plane, it holds \sum_{i=1}^n{f(Tx_n)} = 0. Prove that f = 0



elliptical functions and manifolds

Thu, 22 Sep 2005 04:06:59 GMT

Hello, folks. I hope that I'm in the right group to be asking this. I am no mathematician, although I do know somewhat more than the average bear and can read an equation.

Could someone *please* explain to me in a "dummies" sort of way what an elliptical function is? And do I have it right that a "manifold" is an n-dimensional version of same?

Thanks in advance for either clarifying this for me, or for redirecting me to the right group.




Mon, 07 Jun 2004 18:24:31 GMT

Hey all, I just joined the community. I'm Doug, an undergrad at Dartmouth College. I've spent most of time on physics instead of math, but next year I'm taking algebra and then in the spring I get a chance to study geometric algebra or twistors (one of the two topics) which is exciting. I suppose that's enough of an introduction, these things are generally doomed to be boring anyway.



Travels to Tennessee ...

Tue, 21 May 2002 01:44:53 GMT

I'm at the 6th International Conference on Clifford Algebras and their Applications. Here are some photos from the event.

http://scasey1960.fotki.com/clifford_algebra/


More later. I'm trying to get audio files recorded but my cheap MP3 player doesn't do a very good job in large rooms!



Article on Octonions (with connections to Clifford Algebras)

Fri, 12 Apr 2002 15:55:22 GMT

This month's lead article in Bulletin (New Series) of the American Mathematical Society is a 60-page article by John Baez on Octonions.








Sun, 10 Feb 2002 18:47:21 GMT

Every Friday after work, a mathematician goes down to the bar, sits in the second-to-last seat, turns to the last seat, which is empty, and asks a girl who isn't there if he can buy her a drink.

The bartender, who is used to weird university types, always shrugs but keeps quiet. But when Valentine's Day arrives, and the mathematician makes a particularly heart-wrenching plea into empty space, curiosity gets the better of the bartender, and he says,

"I apologize for my stupid questions, but surely you know there is NEVER a woman sitting in that last stool, man. Why do you persist in asking out empty space?"

The mathematician replies, "Well, according to quantum physics, empty space is never truly empty. Virtual particles come into existance and vanish all the time. You never know when the proper wave function will collapse and a girl might suddenly appear there."

The bartender raises his eyebrows. "Really? Interesting. But couldn't you just ask one of the girls who comes here every Friday if you could buy HER a drink? Never know... she might say yes."

The mathematician laughs. "Yeah, right -- how likely is THAT to happen?"



Anti-gravity??

Thu, 29 Nov 2001 23:17:53 GMT

December is the month for reading. Have people made progress on the first article for this group. I've run across a few books which work through using Clifford Algebras for E&M and Mechanics. I just ordered one from Amazon.com - it's Causality, Electromagnetic Induction, and Gravitation: A Different Approach to the Theory of Electromagnetic and Gravitational Fields, 2nd edition by Oleg D. Jefimenko.

The book's summary includes: One of the most important consequences of the causal gravitational equations is that a gravitational interaction between two bodies involves not one force (as in Newton's theory) but as many as five different forces corresponding to the five terms in the two retarded gravitational and cogravitational field integrals. These forces depend not only on the masses and separation of the interacting bodies, but also on their velocity and acceleration and even on the rate of change of their masses. A series of illustrative examples on the calculation of these new forces is provided and a graphical representation of these forces is given. The book concludes with a discussion of the possibility of antigravitation as a consequence of the negative equivalent mass of the gravitational field energy

Is this a bunch of 'hooey' or what?? The additional effects of gravitation should be observable in several high-mass astronomy examples (neutron star orbits, black hole accretion, et cetera). Does this help to explain the missing mass problem?

Comments??






Great!!

Wed, 31 Oct 2001 18:58:05 GMT

Thanks Pat. Things have been pretty busy here with my quest for a new apartment. I'm glad you've greated this group.



Reading material of interest....

Wed, 31 Oct 2001 17:08:09 GMT

So, what books out there best present Geometric Algebras?

I first discovered them from Pertti Lounesto's book
Clifford Algebras and Spinors

I love this book. The exercises in it are really useful in developing
a comfort with the algebras. They also make a point of demonstrating
some of the common stumbling points that the author has seen repeated
over and over in published papers.



Welcome...

Wed, 31 Oct 2001 16:25:58 GMT

This livejournal evolved out of the Yahoo!Club sofiabookclub.

The suggestion there was to start with Hestenes' paper on Geometric
Algebra:
http://modelingnts.la.asu.edu/pdf/NFMPchapt1.pdf

That paper is a good introduction to Geometric Algebras. Let's discuss.