Pyritohedral icosahedron

The pyritohedral icosahedron, pyrike, or snub truncated octahedron is a convex isogonal polyhedron that is a variant of the icosahedron with pyritohedral symmetry. It has 8 equilateral triangles and 12 isosceles triangles for faces.

It can generally be formed by alternating a semi-uniform truncated octahedron.

This polyhedron can be formed as the hull of three orthogonal rectangles. The short edges of these rectangles then become the set of 6 edges of this polyhedron. If the rectangles have short edges of leength a and long edges (now internal to the polyhedron) of length b, the remaining 24 edges of the polyhedron have length $$\sqrt{\frac{a^2+b^2-ab}{2}}$$. In particular if the rectangles are golden rectangles (that is, b is $$\frac{1+\sqrt5}{2}$$ times greater than a) it gives the regular icosahedron.

A particular case of this polyhedron, where the 24 edges of the equilateral triangles have length $$\sqrt3$$ and the remaining 6 edges have length $$\sqrt2$$, appears as the alternation of the uniform truncated octahedron.

Another case of this polyhedron, with 6 unit edges and 24 of length $$\frac{1+\sqrt5}{2}$$, can be obtained by removing the 8 vertices of an inscribed cube from a regular dodecahedron.