Perfect Shapes in Higher Dimensions: Polytopes and Coxeter Groups скачать в хорошем качестве

Perfect Shapes in Higher Dimensions: Polytopes and Coxeter Groups

We describe regular polygons, regular polyhedra like the platonic solids, and the higher dimensional regular polytopes like the tesseract, 24 cell and 120 cell. We describe the structure of all regular polytopes and how they can be studied intuitively, using coordinates, Schläfli symbols and in terms of symmetry. In particular we describe how reflections and Coxeter groups can be used to understand all the symmetries of regular polytopes. Basically the idea is to use generalized kaleidoscopes. We describe how this relates to Coxeter groups and Coxeter diagrams before moving over to connecting these ideas with those of root lattices and Dynkin diagrams. We study the same phenomena in terms of root lattices. When we focus on Coxeter diagrams which are ADE graphs (i.e., of the form An, Dn or En) we find that the resulting phenomena (i.e., the resulting geometry and reflection groups/ Coxeter groups) can easily be studied in terms of ADE graphs. We study the An lattice using geometry, practical examples and models. We study the Dn lattice similarly. We also describe how these structures can easily be described in terms of Wildberger’s mutation game. 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)