A Pyritohedral and Pentagon-Faced Polyhedron

pyritohedral 36 pentagons

Twelve of the faces of this polyhedron are pink, and the other twenty-four are blue. It has no faces which are not pentagons. I made it using Stella 4d: Polyhedron Navigator, which is avialable at http://www.software3d.com/Stella.php

Two Compounds with Pyritohedral Symmetry: the Icosidodecahedron / Truncated Octahedron Compound, and the Rhombic Triacontahedron / Tetrakis Cube Compound

Compound of Icosidodeca and Trunc Octa its pyritohedralCompound of RTC and tetrakis cube its pyritohedral

Stella 4d, a program you can try here, was used to create these two compounds. Both have pyritohedral symmetry: the symmetry of a standard volleyball. The two compounds are also duals.

On Icosahedra, and Pyritohedral Symmetry

Icosa pyrito & two tets

In this icosahedron, the four blue faces are positioned in such a way as to demonstrate tetrahedral symmetry. The same is true of the four red faces. The remaining twelve faces demonstrate pyritohedral symmetry, which is much less well-known. It was these twelve faces that I once distorted to form what I named the “golden icosahedron” (right here: https://robertlovespi.wordpress.com/2013/02/08/the-golden-icosahedron/), but, at that point, I had not yet learned the term for this unusual symmetry-type.

To most people, the most familiar object with pyritohedral symmetry is a volleyball. Here is a diagram of a volleyball’s seams, found on Wikipedia.


Besides the golden icosahedron I found, back in 2013, there is another, better-known, alteration of the icosahedron which has pyritohedral symmetry, and it is called Jessen’s icosahedron. Here’s what it looks like, in this image, which I found at http://en.wikipedia.org/wiki/Jessen%27s_icosahedron.


The rotating icosahedron at the top of this post was made using Stella 4d, a program which may be purchased, or tried for free (as a trial version) at http://www.software3d.com/Stella.php.