Description

In two dimensions there is a regular polygon with n sides for every n>2, but on the other hand, you can only find a regular tessellation of the plane using certain regular polygons. In three dimensions, there are only five Platonic solids (polyhedra with just one shape of face), and you can see the tesselations of the plane as a sort of 'limiting' case of three dimensional polyhedral—the limit where the polyhedron never quite gets round to curving up to meet itself. This project explores the analogous ideas in higher dimensions, in particular looks at 4 dimensional Platonic solids, and regular tesselations or other decorations of Euclidean space of dimension greater than 2.

Prerequisites

Codes and Geometric Topology - MATH2141 would provide a good introduction.

Resources

There are many good sources for material on convex geometry in low dimensions, but a really nice book with a lot of interesting insights and related topics is "The Symmetries of Things" by John Conway, Heidi Burgiel and Chaim Goodman-Strauss. See also the classic text "Tilings and Patterns: An Introduction" by Branko Grunbaum and G. C. Shephard.