# Grünbaum polyhedron

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Grünbaum polyhedron | |
---|---|

Rank | 3 |

Type | Isogonal |

Space | Spherical |

Elements | |

Faces | 8+8 equilateral triangles, 24+24 scalene triangles |

Edges | 24+24+48 |

Vertices | 24 |

Measures | |

Central density | 0 |

Related polytopes | |

Army | Snub cube |

Convex hull | non-uniform snub cube |

Abstract & topological properties | |

Flag count | 384 |

Euler characteristic | –8 |

Schläfli type | {3,8} |

Orientable | Yes |

Genus | 5 |

Properties | |

Symmetry | B_{3}+, order 24 |

Convex | No |

The **Grünbaum polyhedron** is a realization of the Fricke-Klein map in 3-dimensional Euclidean space with the maximum possible symmetry. The Fricke-Klein map is regular, and thus the Grünbaum polyhedron is regular under its automorphism group, however it is not regular under its symmetry group.

## Construction[edit | edit source]

The Grünbaum polyhedron can be constructed by blending two non-identical snub cubes with coincident vertices. The inner snub cube is the same as the outer snub cube with each of its square faces rotated about its own center a quarter turn. If a uniform snub cube is used the inner snub cube will self-intersect. Typically a non-uniform version is used to avoid this.

## Gallery[edit | edit source]

## External links[edit | edit source]

- Hartley, Michael. "{3,8}*384".