I always just assumed the link between algebra and geometry was pretty solid, but actually it's two different worlds, similar to Poincare's distinction between logical and intuitive that I've discussed elsewhere (and so has Terence Tao). So it's nice to find that someone found an 
elegant synthesis between the two worlds. Apparently it's so elegant it compresses Maxwell's first four equations into a single one, for example. Why isn't this already well known? Curious? Me, too. So here's a lonnnnng page to get you started. Me too, when I find the time to read it all, which is why this post is here.
Clifford Algebra, a.k.a. Geometric Algebra, is a most extraordinary synergistic confluence of a diverse range of specialized mathematical fields, each with its own methods and formalisms, all of which find a single unified formalism under Clifford Algebra. It is a unifying language for mathematics, and a revealing language for physics.