Symmetry groups of regular polytopes in three and four dimensions

Abstract

A pentagon is an example of a highly symmetric polygon in two-dimensional space. The three-and four-dimensional analogue of these polygons are the regular polyhedra and the regular polytopes. There exist five regular polyhedra in three-dimensional space and these are called the Platonic solids. These five Platonic solids are the tetrahedron, cube, octahedron, dodecahedron and the icosahedron. In four-dimensional space, the regular polytopes are the 5-cell, the 8-cell, also called the tesseract, the 16-cell, the 24-cell, the 120-cell and the 600-cell. The main focus in this thesis is to describe the symmetry group of the icosahedron, to introduce the Icosians, which are related to the rotation group of the icosahedron, and to study the action of the symmetries of the 600-cell on the twenty-five 24-cells it circumscribes. Firstly, the symmetry groups of the Platonic Solids, the regular polytopes in three dimensional space are established. Then it will be shown that there eixsts a two-to-one map from the the group H₁ of unit quaternions to the group SO(3), the group of 3 x 3 orientation-preserving matrices. This map will be used to describe the binary groups, which are double covers of the rotation groups of the Platonic solids. After that, the symmetry group of the tesseract will be studied both via an isomorphism between G := {± 1 }⁴ x S₄} and the symmetry group of the tesseract as well as geometrically via rotation planes. Then, the 24-cell and the 600-cell will be defined as the four-dimensional regular polytopes whose vertices are the quaternions from the binary tetrahedral group and the binary icosahedral group, the Icosians. It will be shown that twenty-five 24-cells inscribe a 600-cell and that there are 10 ways to decompose the vertices of a 600-cell into the vertices of 5 disjoint 24-cells. Next, it will be shown that these 10 decompositions are chiral, 5 being 'left-handed' and 5 being 'right-handed'. Finally, it is shown that the symmetry group of the 600-cell acts on these 5+5 decompositions by permutation, each permutation being described by an element from A₅ x A₅ ⋊ {±1}, where -1 acts on A₅x A₅ by interchanging the factors of A₅x A_5.