# Pyritohedral icosahedron

Pyritohedral icosahedron
Rank3
TypeIsogonal
Notation
Bowers style acronymPyrike
Coxeter diagramo4s3s ()
Elements
Faces12 isosceles triangles, 8 triangles
Edges6+24
Vertices12
Vertex figureMirror-symmetric pentagon
Related polytopes
ArmyPyrike
RegimentPyrike
DualPyritohedron
ConjugatePyritohedral great icosahedron
Abstract & topological properties
Flag count120
OrientableYes
Properties
SymmetryB3/2, order 24
Flag orbits5
NatureTame

A pyritohedral icosahedron, pyrike, or snub truncated octahedron is an isogonal icosahedron with pyritohedral symmetry. It has 8 equilateral triangles and 12 isosceles triangles for faces.

The regular icosahedron is a convex special case where all edges are equal, resulting in full icosahedral symmetry. Jessen's icosahedron is a nonconvex special case with some interesting properties, namely that all dihedral angles are 90 degrees and it is an infinitesimally flexible polyhedron.

The convex forms of this polyhedron can be formed by alternating a semi-uniform truncated octahedron, or as the convex 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 ${\displaystyle {\frac {1+{\sqrt {5}}}{2}}}$ 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 ${\displaystyle {\sqrt {3}}}$ and the remaining 6 edges have length ${\displaystyle {\sqrt {2}}}$, appears as the alternation of the uniform truncated octahedron.

Another case of this polyhedron, with 6 unit edges and 24 of length ${\displaystyle {\frac {1+{\sqrt {5}}}{2}}}$, can be obtained by removing the 8 vertices of an inscribed cube from a regular dodecahedron.