Very interesting! But it requires really a skilled lecturer to make students understand the concepts. I've heard advanced mathematical physics by a, well, bad lecturer, who taught everything on some abstract level which I never heard from previously (and after). I had to learn differential geometry and all that later on my own in order to graduate in some topic in general relativity.
Various arguments have been proposed for adopting this model such as, conformal space:
• is homogeneous and consequently removes the origin as being a special point
• supports points and lines at infinity
• provides a single geometric mechanism for representing lines, circles, planes and spheres, which introduces a rare quality of elegance to problem solving
• enjoys all the normal features of Euclidean space.
A model of space that possesses such a range of positive features sounds too good to be true! But remember, in mathematics there is no such thing as a free lunch. So what price must we pay for this model?
Uh what?! So it actually costs money to use... That rather put me off Hestenes' noble mission. It seems...bizarre that what seems a whole subfield of mathematics is patented, rather different to patenting a particular algorithm or technique. Very strange, after reading so much of Hestenes' wondering why the superior GA isn't used everywhere. (Is it to make the world a better place..or to make Hestenes rich?..) I stopped reading at that point.
As far as using Geometric Algebra in a Euclidean setting goes, most of my programming is in 2D, but when I do 3D next I will consider using it, it seems very cool. Most of the features/differences/advantages over the usual vector/matrix geometry only appear in 3 or more dimensions. Vince's book is great at explaining it, highly recommended.
As a result, all kinds of pseudoscience has gotten "patented", for example this patent for detecting "tachyonized" materials (you know, materials that have been infused with tachyons, the hypothetical particles that travel faster than light):
So this patent very likely wouldn't hold up in court. It was approved in a time when the USPTO more or less gave up on adjudicating patentability (thanks to the State St. decision).
I like to mention on such threads:
"Geometric Algebra for Electrical and Electronic Engineers"
> In this paper, we explicate the suggested benefits of Clifford's geometric algebra (GA) when applied to the field of electrical engineering. Engineers are always interested in keeping formulas as simple or compact as possible, and we illustrate that geometric algebra does provide such a simplified representation in many cases. We also demonstrate an additional structural check provided by GA for formulas in addition to the usual checking of physical dimensions. Naturally, there is an initial learning curve when applying a new method, but it appears to be worth the effort, as we show significantly simplified formulas, greater intuition, and improved problem solving in many cases.
And then claims that aerospace engineers mostly use quaternions. It's from 2002, so it's sense of what's in vogue might be dated, but I'd understood at least in the graphics manipulation world, the vector/matrix methods were dominant. Those are separate industries, so both statements could be true, anybody have opinions?
Some 10 pages in I realized it's explaining the concepts I've internalized years ago in a way I can't comprehend at all.
Doing electrodynamics without using alternating products and differential forms feels so wrong...
The fact that the same physical system can be described by very different kinds of mathematics is often very illuminating and interesting, rather than confusing. It's one of the reasons that physics can inspire mathematics, rather than maths just being a tool that physicists use. Different types of mathematics often allow you to emphasise different properties of the same system, so being comfortable in more than one language brings you freedom and new insights.
Again, I don't use any of this in real life, so YMMV, but it looks very clean, fast, and intuitive compared to (waves at everything else).