
Thursday, June 11th, 2009

3:42 pm  Orthogonal Transformations with Complex Numbers, Quaternions, and Clifford algebras

patrickwonders

Crossposted from 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 reread 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 3space to Clifford algebras again in a
series of posts on my
website.
I have posted four articles in that series so far:

(comment on this)

 Monday, June 8th, 2009

12:07 pm  Plane Transformations with Complex Numbers and Clifford Algebras

patrickwonders

crossposted to 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 reread 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 3space to Clifford algebras again in a
series of posts on my
website.
I have posted two articles in that series so
far:

(comment on this)

 Saturday, March 11th, 2006

1:46 pm  Dead community!

 Wednesday, January 18th, 2006

1:24 pm

sethx

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

(comment on this)

 Wednesday, September 21st, 2005

9:04 pm  elliptical functions and manifolds

not_fade_away

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 ndimensional version of same?
Thanks in advance for either clarifying this for me, or for redirecting me to the right group.

(1 comment  comment on this)

 Monday, June 7th, 2004

2:13 pm

doogly

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.

(4 comments  comment on this)

 Monday, May 20th, 2002

6:44 pm  Travels to Tennessee ...

 Friday, April 12th, 2002

11:55 am  Article on Octonions (with connections to Clifford Algebras)

patrickwonders

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

(1 comment  comment on this)

 Tuesday, February 19th, 2002

4:52 pm  Conference ...in case you are interested

 Sunday, February 10th, 2002

12:48 pm

kvschwartz

Every Friday after work, a mathematician goes down to the bar, sits in the secondtolast 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 heartwrenching 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?"

(2 comments  comment on this)

 Thursday, November 29th, 2001

3:11 pm  Antigravity??

scasey1960

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 highmass astronomy examples (neutron star orbits, black hole accretion, et cetera). Does this help to explain the missing mass problem?
Comments??

(3 comments  comment on this)

 Friday, November 9th, 2001

12:12 pm  So, things are pretty quiet here.... how about a poll question...

patrickwonders

Have you played with Geometric Algebras or Clifford Algebras
before?
What interests you about Geometric Algebras?
The applications to quantum mechanics
0(0.0%)
The sheer algebra of it
2(25.0%)
The noncommutativity
0(0.0%)
Applications of their subalgebras
0(0.0%)
Potential future applications
0(0.0%)
current mood: curious

(comment on this)

 Wednesday, October 31st, 2001

11:00 am  Great!!

scasey1960

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

(comment on this)


12:08 pm  Reading material of interest....

patrickwonders

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.
current mood: grateful

(comment on this)


11:26 am  Welcome...




