The Code Behind Geometry : Coxeter Groups

We explore the thesis that Coxeter groups underlie geometry and fundamental physics. In particular we focus upon how Coxeter groups can be used to generate regular polytopes and honeycombs. We discuss how this can be achieved using the Wythoff construction, as described in Chapter 11 of Coxeter's book "Regular Polytopes". We investigate this construction from several perspectives, using matrices, representation theory and group theory. This is all closely related to my other video Perfect Shapes in Higher Dimensions: Polytopes and Coxeter Groups https://www.youtube.com/watch?v=AbFAWNb0U3k&list=PLCTMeyjMKRkr6Po3BHAbgXnVtXbeCKdOB&index=6 Resources An Exceptionally Simple Theory of Everything A. Garrett Lisi https://arxiv.org/abs/0711.0770 In terms of a simple game called the "mutation game", which is played on a graph. This changes the populations, which we obtain descriptions of from Wildberger The Ubiquity of ADE Graphs, and the Mutation and Numbers Games | Math Seminars | NJ Wildberger • The Ubiquity of ADE Graphs, and the Mutati... • Exceptional Structures in Mathematics and ... Wildberger's mutation game gives an elementary way to understand the ADE lattices without requiring any prior math skills. The following texts where very useful for preparing this video and highly recommended: Regular Polytopes H.S.M. Coxeter’s COXETER GROUPS Gert Heckman https://www.math.ru.nl/~heckman/Coxet... From the Icosahedron to E8 John C. Baez https://arxiv.org/abs/1712.06436 Also the following videos were also helpful Norman Wildberger The Mutation Game | Dynamics on Graph 1 in Explore Research Level Maths | Wild Egg Maths • The Mutation Game | Dynamics on Graphs and... John Baez This Week's Finds 4: Coxeter and Dynkin diagrams • This Week's Finds 4: Coxeter and Dynkin di... (Also see This Week's Finds 5 and This Week's Finds 6 in the same playlist)