# W'

W' | |
---|---|

Rank | 3 |

Type | Acrohedron |

Space | Spherical |

Notation | |

Stewart notation | W' |

Elements | |

Faces | 1 hexagon, 3 squares, 3 pentagons, 6+3+3+1 triangles |

Edges | 3+3+3+3+6+6+6+6 |

Vertices | 3+3+6+6 |

Measures (edge length 1) | |

Volume | |

Central density | 1 |

Abstract & topological properties | |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | A_{2}×I, order 6 |

Convex | Yes |

Nature | Wild |

**W'** is a convex^{[note 1]} polyhedron with all regular faces. However it is not a Johnson solid, since several of its faces are coplanar. It is notable for being the first discovered example of a 6-5-4 acrohedron. It was discovered by John Horton Conway and given its name by Bonnie Stewart.

## Vertex coordinates[edit | edit source]

A W' of edge length 1 has vertices given by the following coordinates:

- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- .

## Related polytopes[edit | edit source]

W' is closely related to the triangular hebesphenorotunda, a Johnson solid. The two have the same face counts, symmetry and volume.^{[1]}

Its coplanar faces are resolvable by excavating them with four tetrahedra, resulting in a 28-faced 6-5-4 acrohedron. Stewart names this polyhedron **W''**. W'' is weakly quasi-convex.^{[1]}

Alex Doskey found that W' can be augmented with a square pyramid to produce a 6-5-3-3 acrohedron.

W' is non-self-intersecting. Richard Klitzing found a self-intersecting 6-5-4 acrohedron with the same symmetry, that has only 17 faces.

## External links[edit | edit source]

- Jim McNeill. "6-5-4."

## Notes[edit | edit source]

- ↑ If the definition that a convex polytope is the convex hull of its vertices is used, W' is not convex. However the interior of W' is a convex set, so it satisfies the interior definition. Stewart considered W' to be convex.

## References[edit | edit source]

- ↑
^{1.0}^{1.1}Stewart (1964:168)

## Bibliography[edit | edit source]

- Stewart, Bonnie (1964).
*Adventures Amoung the Toroids*(2 ed.). ISBN 0686-119 36-3.