Constructing Polyhedra Using the Icosahedral Group

polyhedra

RhombicHexecontahedron

faces to show

60

An equilateral triangle can be rotated onto itself. The cyclic group

C

3

describes such actions. Adding mirror images gives the dihedral group

D

3

. 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,

T

(tetrahedral group, order 12),

O

(octahedral group, order 24), and

I

(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

I

as an order 60 set of rotation matrices, and applies these transformations to an appropriately chosen polygon to generate various 60-sided polyhedra.