I'll attend Handmade Cities - Boston next week. One of the speakers is giving a talk on a subject that I'm very interested in: Geometric Algebra.

A few years ago I was struggling to get an intuition for the math that describes rotation and motion in 3D. The standard ways of handling rotation involves linear algebra, 3x3/4x4 matrices and/or quaternions, but the explanations were very unsatisfying. There were different rules for manipulating points, lines, and planes, and the techniques didn't generalize to higher dimensions.

My curiousity led me to studying Geometric Algebra (GA). It's a beautiful subject that unifies several seemingly-different number systems into one coherent whole. GA is not taught in mainstream academia, but its community believes (and I agree) that it should replace several parts the standard math curriculum for STEM. GA has been gaining popularity in recent years, with talks given at SIGGRAPH (computer graphics conference), GDC (game developer conference) and in academic conferences.

I was specifically interested in Charles Gunn's Projective Geometric Algebra, or PGA for short. If GA was non-mainstream, then PGA was doubly so! The GA community mostly used techniques from Conformal Geometric Algebra (CGA) and seemed to ignore PGA.

This week I learned that there's been a lot of progression in PGA in just the last 2-3 years. The PGA moniker now refers to Plane-based Geometric Algebra, and the community has come to accept Charles Gunn's approach with more enthusiasm:

It was the unconventional nature of duality in PGA that hampered its acceptance by the GA community...

[The] community held onto CGA, and Gunn's message was not heard. His admonition that we confused 'duality' with 'polarity'... was considered too mathematically prissy.

source

I'm very interested in representations of thought¹, and GA is full of better representations than the ones that are in the mainstream!