# Geometric Algebra

## Main Question or Discussion Point

I've seen a number of books and articles touting Geometric Algebra as an important new area of math that will have large application to physics. Is there anything to these claims? Is it worth studying for a physics student?

phyzguy
I think it is definitely worth studying, and I strongly recommend this text:

"Geometric Algebra for Physicists" by Doran and Lasenby.

EWH
Sorry to bump up this old thread. Granpa's got some good references to look at, here are some more pasted in from an email I wrote a while back:

Geometric Algebra - not easy, but worth it. There's not much that can't be done with GA - all physics (including relativity and QM) Maxwell's equations in relativistic form condense to 4 symbols. Angles, solid angles and scalars are cleanly differentiated as being different dimensions. Quaternions and complex math come along free for the ride. It works in any dimension and any signature. Interval arithmetic is just a special 2+1-D version of a conformal/Minkowski space. It's fast. It's intuitive. (sorta) It's coordinate-free. It slices it dices! (Well, it does have a lot of blades, anyway.)

For introductory tutorials with both PDF and interactive graphical/command-line GA calculator program (GA Viewer) tutorials:
http://www.science.uva.nl/ga/tutorials/index.html [Broken] (The conformal model is in 5-D so you'll want some introduction.)

A good basic reference and primer: http://www.jaapsuter.com/geometric-algebra.pdf

For a good book: Geometric Algebra For Computer Science, An Object Oriented Approach to Geometry,
http://www.geometricalgebra.net/ [Broken] (official site)
http://www.scribd.com/doc/7149305/Geometric-Algebra-for-Computer-Science-an-ObjectOriented-Approach-to-Geometry [Broken] (online version for preview - the paper 2nd edition is better, and the cheat-sheets in the cover pages are very useful.)

That's more than enough, but there are many other good resources such as David Hestenes' GA page, the Cambridge GA group (they founded a successful company a few years ago to do real-time radiosity lighting and physics for games - Geomerics), Ian G.C. Bell's "Maths for (Games) Programmers" (needs an old version of Netscape to view, there's a link on his site to such versions. Bell co-wrote the first 3-D (and space-trading) game for personal computers, Elite, but retired to do fractal body paint on pretty girls at raves.)​

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There is also this, for someone with some basic prior knowledge of tensor products of vector spaces:

Clifford algebra, geometric algebra, and applications
Douglas Lundholm, Lars Svensson

http://arxiv.org/abs/0907.5356v1

(I have only just started reading it a couple days ago.)

It slices it dices! (Well, it does have a lot of blades, anyway.)
Chuckle

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jbunniii
Homework Helper
Gold Member
While browsing on Amazon recently, I came across this title: Linear and Geometric Algebra. Based on the "search inside" feature, it looks good and very accessible, not to mention reasonably priced. Has anyone here read it?

EWH
No, but I read his GA paper a few years back and recall it as pretty good. I'd look at the the primer from Jaap Suter first, and try out the GAViewer software and its tutorials before spending money on books. (links in my previous post) If you're already into linear algebra, then McDonald's book might be the best first book, but take a look at "Geometric Algebra For Computer Science", too.

EWH
Sankaku: Yes, that's good for the real mathy-math types, not the place to start unless you eat abstract algebra for breakfast, lunch and dinner.

Even if you aren't that hardcore (and I'm not), the Wikipedia article: "Classification of Clifford Algebras" is worth a look, it ties a lot of different things together - Bott periodicity, classes of square matrices with real, complex, quaternion, split-complex, and split-biquaternion elements, all the possible signatures ... just 5 screen-fulls of some of the densest information in the universe.

I have tried to tease out some of the basic implications in the attached spreadsheet, no implied warranties, it's basically scratch-work for my own study, but perhaps others may get some use out of it.

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While browsing on Amazon recently, I came across this title: Linear and Geometric Algebra. Based on the "search inside" feature, it looks good and very accessible, not to mention reasonably priced. Has anyone here read it?
Yes, I have it. I have some mixed feelings about it. The Linear Algebra part is a reasonable review if you already know the material. I would not use it as the only book for a first course, though.

The GA part is a bit unsatisfying to me, as a mathematician. It is reasonable as a quick introduction, but does not give more than superficial intuition. It also doesn't give enough rigour (for me) and takes an approach I would (perhaps unfairly) call "application oriented." In other words, I hear something like this in the back of my head: "Hey physics and engineering people, here is a cool tool! We will show you how to use it but not show you all the machinery that makes it tick."

More charitably, the book seems like decent lecture notes to supplement other material. I think the paper that EWH linked to is a bit quicker and more to the point. I appreciate MacDonald's hard work at making it all accessible, so don't take my criticisms too seriously. Some time, when I have it figured out properly, I will have to write the missing introduction aimed at the (very small) demographic that includes me right now...

Sankaku: Yes, that's good for the real mathy-math types, not the place to start unless you eat abstract algebra for breakfast, lunch and dinner.
Well, lunch is ok. I am still working on breakfast.

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I have to echo that sentiment. I had hoped the book would turn out to be a comprehensive rewriting of linear algebra in terms of matrix-free notation, giving new insights into the ideas of the subject. I found the geometric algebra aspects quite basic, though.

Rewriting linear algebra in the language of operators on vectors instead of matrices and components is incredibly powerful and clarifying. Operations like the trace and determinant can be written in entirely coordinate-free forms, as well as a formula for calculating the adjoint (transpose) and inverse of a linear operator--and all without invoking a matrix representation at any time. I didn't feel like the book fully explored that power.

Too bad, too. I really liked MacDonald's paper that derives the GA framework from just basis vectors and the geometric product.

GA Notes and Software

If you are interested I have extensive geometric algebra/calculus notes (based on Doran and Lasenby) and symbolic software (python) at

https://github.com/brombo/GA [Broken]

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hunt_mat
Homework Helper
No, but I read his GA paper a few years back and recall it as pretty good. I'd look at the the primer from Jaap Suter first, and try out the GAViewer software and its tutorials before spending money on books. (links in my previous post) If you're already into linear algebra, then McDonald's book might be the best first book, but take a look at "Geometric Algebra For Computer Science", too.
I've just started a postdoc in geometric algebra and I am reading the book, "Geometric algebra for computer scientists" and I am finding it VERY good, it has a great deal of supplementary material and is actually very good. My one caveat though, is that you should be reasonably familiar and happy with linear algebra first. It really has a great deal of things going for it.

My main personal interest(as opposed to the requirements of my postdoc) is applying geometric calculus to fluid dynamics.