# Geometric algebra

By Marcus Wilson 16/09/2009 1

Here’s a lovely quote that students will empathize with:

“A recent study on the use of vectors by introductory physics students summarized the conclusions in two words: “vector avoidance”. This state of mind tends to propagate through the physics curriculum. In some 25 years of graduate physics teaching, I have noted that perhaps a third of the students seem incapable of reasoning with vectors as abstract elements of a linear space…I have come to regard this concept of a vector as a kind of conceptual virus, because it impedes development of a more general and powerful concept of a vector…”

David Hestenes, Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics. Reference made to E. Redish and G. Shama, AAPT Announcer vol 27, p98 (1997)

Very, very true. But help is at hand, in the form of geometric algebra. It is not unrelated to the complex algebra I’ve talked about, but has an advantage in that it is all firmly based in reality. Geometry is real. Objects are three dimensional things, with volume, surfaces, edges and vertices (corners). We should (and do) have a method of dealing with all these entities properly.

Geometric algebra has been around for a while, but popularised recently by the likes of David Hestenes and Anthony Lasenby, amongst others. I’m searching the net for a really simple introduction to it (something suitable for a blog) but haven’t got there yet. I would call it ‘vectors done properly’. As a physicist, I find geometric algebra a simpler and way more powerful method of dealing with geometrical objects such as vectors (and these come up everywhere in physics) than the more traditional approaches taught at school and university. So why isn’t it taught?

### One Response to “Geometric algebra”

• “It is not unrelated to the complex algebra …” In fact, complex numbers are easily found to be a sub-algebra of geometric algebra. Quarternions also fall nicely in line, being nothing more that the bi-vector basis – oriented planar quantities – of 3D geometric algebra.

Complex numbers are to geometric algebra, what checkers are to chess. Working with the multi-dimensional analogues of what we typically call the imaginary i reveals a structure to space that makes cosmic sense of many otherwise arcane concepts, many of which turn out to be flat-out confused.

Pauli spin matrices are simply wrong-headed in their approach to vector and matrix notation, neither of which turn out to be in the least bit necessary.

There’s more, lots more. Alan Macdonald has a very good web site as well as a recently published under-grad covering aspects of both linear and geometric algebra. That website has links to many of the important introductory papers. You should visit. Here’s the site:

http://faculty.luther.edu/~macdonal/