Constructing Polyhedra Using the Icosahedral Group

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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.


Contributed by: Izidor Hafner (September 2011)
Additional code by Ed Pegg Jr
Open content licensed under CC BY-NC-SA



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.