Pyritohedral icosahedron

Pyritohedral icosahedron
Rank	3
Type	Isogonal
Space	Spherical
Notation
Bowers style acronym	Pyrike
Coxeter diagram	o4s3s
Elements
Faces	12 isosceles triangles, 8 triangles
Edges	6+24
Vertices	12
Vertex figure	Mirror-symmetric pentagon
Measures (edge length 1)
Central density	1
Related polytopes
Army	Pyrike
Regiment	Pyrike
Dual	Pyritohedron
Conjugate	Pyritohedral great icosahedron
Abstract properties
Euler characteristic	2
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B3/2, order 24
Convex	Yes
Nature	Tame

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 length 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.

External links

• Klitzing, Richard. "snit".