Can anyone recommend a good book (or reading list) that would allow me to transition from my simplistic Undergrad Physics-based view of Geometry to being able to understand the Maths-based world of Geometry, and the Geometric relationship with Clifford Algebras?
Got to love it! After years of trying and bouncing off, then finally asking the question, I finally figured it out: The missing link I was looking for was the subject of Abstract Algebra. It explains, Groups, Rings, and the Geometric side of it expands into Vector Spaces. Exactly the correspondance that I was looking for...
I can now complete my roadmap that will allow me to progress my physics-based understanding of 3D geometry to a level where I am able to understand (and maybe one day contribute to) the discussions in this wonderful thread!
I studied the regular 3D Geometric Algebra of Grassman/Hestenes about 10 - 20 years ago and it immediately answered all my questions about the link between Matrices, Vectors, and simple Geometry that nobody at University could actually explain. (The professors seemed to get quite testy when I asked them to explain why the relationship between Vectors and Matrices was what it was... That said, they always got testy whenever I asked them the whys of anything!)
Anyway I realized that the properties of Geometric Algebra could explain a lot more than the link between Vectors and Matrices, but for it to be able to explain Wave Particle Duality, Time, and a link between Mass, Charge, and the Lorentz transform, a lot more dimensions were needed than just our regular 3. (I never accepted Minkowski Spacetime as real, and have yet to see anything that can convince me that Time is an actual dimension any more than Frequency/Wavelength, the population of rabbits in a field, or any other independent degree of freedom is...)
Sorry about the distraction!