# Pyritohedral icosahedron

Pyritohedral icosahedron
Rank3
TypeIsogonal
SpaceSpherical
Notation
Bowers style acronymPyrike
Coxeter diagramo4s3s
Elements
Faces12 isosceles triangles, 8 triangles
Edges6+24
Vertices12
Vertex figureMirror-symmetric pentagon
Measures (edge length 1)
Central density1
Related polytopes
ArmyPyrike
RegimentPyrike
DualPyritohedron
ConjugatePyritohedral great icosahedron
Abstract properties
Euler characteristic2
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB3/2, order 24
ConvexYes
NatureTame

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 ${\displaystyle \sqrt{\frac{a^2+b^2-ab}{2}}}$. In particular if the rectangles are golden rectangles (that is, b is ${\displaystyle \frac{1+\sqrt5}{2}}$ 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 ${\displaystyle \sqrt3}$ and the remaining 6 edges have length ${\displaystyle \sqrt2}$, appears as the alternation of the uniform truncated octahedron.

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