# Császár polyhedron

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Császár polyhedron | |
---|---|

Rank | 3 |

Type | Regular toroid, Regular map |

Notation | |

Schläfli symbol | |

Elements | |

Faces | 2+2+2+2+2+2+2 scalene triangles |

Edges | 1+1+1+2+2+2+2+2+2+2+2+2 |

Vertices | 1+2+2+2 |

Vertex figure | 2+2+2 irregular hexagons, 1 bilaterally-symmetric hexagon |

Related polytopes | |

Dual | Szilassi polyhedron |

Abstract & topological properties | |

Flag count | 84 |

Euler characteristic | 0 |

Orientable | Yes |

Genus | 1 |

Skeleton | K_{7} |

Properties | |

Symmetry | K_{2}+×I, order 2 |

Flag orbits | 42 |

Convex | No |

History | |

Discovered by | Ákos Császár |

First discovered | 1949 |

The **Császár polyhedron** (Hungarian: [ˈt͡ʃaːsaːr], approximate English pronunciation *CHA-sar*) is a toroidal polyhedron without any diagonals; that is, every pair of vertices is connected by an edge. It is the dual of the Szilassi polyhedron. It is a regular toroid and regular map, but it is not abstractly regular.

It has 14 scalene triangular faces, 7 vertices, and 21 edges. 6 faces meet at each vertex.

Its vertices and edges are an embedding of the complete graph K_{7} onto a torus of genus 1.

It has the same edge skeleton as the 7-2 step prism.

## Vertex coordinates[edit | edit source]

This polytope is missing vertex coordinates. (April 2024) |

## External links[edit | edit source]

- Wikipedia contributors. "Császár polyhedron".
- McCooey, David. "Csaszar Polyhedron"

- Wedd, N. The dual Heawood map

## Bibliography[edit | edit source]

- Brehm, Ulrich; Kühnel, Wolfgang (2008), "Equivelar maps on the torus",
*European journal of combinatorics*, doi:10.1016/j.ejc.2008.01.010