I recently had a bit of an epiphany around GA the other day (and Dual numbers, and Screw Theory with Twists and Wrenches): I realized that I had it all backwards. I’d seen GA as an alternative formulation for projective geometry and for Maxwell’s equations, Dual Numbers for auto-differentiation, and Screw Theory for rigid-body dynamics; the question that had always bothered me was “how in the hell did they manage to take these engineering problems and derive these crazy linear algebra theories to solve them?”<p>The realization was that it went the other way around! Clifford Algebras are an interesting linear algebra formulation. As it turns out, by appropriately defining the e^2=0, e^2=1, and e^2=-1 axes of the general Clifford Algebra framework, a bunch of these engineering problems <i>map</i> into the algebra and then adopt a super compact notation for solving them.