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.