# 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 | B_{3}/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 . In particular if the rectangles are golden rectangles (that is, b is 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 and the remaining 6 edges have length , appears as the alternation of the uniform truncated octahedron.

Another case of this polyhedron, with 6 unit edges and 24 of length , can be obtained by removing the 8 vertices of an inscribed cube from a regular dodecahedron.

## External links[edit | edit source]

- Klitzing, Richard. "snit".