Constructing Polyhedra Using the Icosahedral Group
An equilateral triangle can be rotated onto itself. The cyclic group describes such actions. Adding mirror images gives the dihedral group . All are subgroups of the infinite special orthogonal group SO(2), also known as the group of 2×2 rotation matrices.
In 3D, SO(3) describes the rotations of a sphere, or the 3×3 rotation matrices (all with determinant 1). There are three finite subgroups, (tetrahedral group, order 12), (octahedral group, order 24), and (icosahedral group, order 60). These describe motions of the given polyhedron onto itself. Each group can be doubled in size with the addition of mirror images.
This Demonstration uses as an order 60 set of rotation matrices, and applies these transformations to an appropriately chosen polygon to generate various 60-sided polyhedra.