# Szilassi polyhedron

Szilassi polyhedron | |
---|---|

Rank | 3 |

Type | Regular toroid |

Space | Spherical |

Elements | |

Faces | 2+2+2 irregular hexagons, 1 bilaterally-symmetric hexagon |

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

Vertices | 2+2+2+2+2+2+2 |

Vertex figure | 2+2+2+2+2+2+2 scalene triangles |

Related polytopes | |

Dual | Császár polyhedron |

Abstract & topological properties | |

Flag count | 84 |

Euler characteristic | 0 |

Orientable | Yes |

Genus | 1 |

Skeleton | Heawood graph |

Properties | |

Symmetry | K_{2}+×I, order 2 |

Convex | No |

History | |

Discovered by | Lajos Szilassi |

First discovered | 1977 |

The **Szilassi polyhedron** (Hungarian: [ˈsilɒʃːi], approximate English pronunciation *SEE-la-shi*) is a toroidal polyhedron. With seven irregular hexagonal faces, it has the least number of faces out of any toroid. It also has the unusual property that each of its faces is adjacent to all of the other faces. It is the dual of the Császár polyhedron.

It has 14 vertices and 21 edges. Three faces meet at each vertex. Among the faces, there are three pairs of identical faces and one unique face. The unique face is the only convex face among the seven.

It is an example of a regular toroid. It is not a regular polyhedron in the traditional sense, but as an abstract polyhedron it is weakly regular, meaning that it is abstractly vertex-transitive, edge-transitive, and face-transitive.

The only other known polyhedron with all of its faces adjacent to all of its other faces is the tetrahedron. It is theoretically possible to embed in 3D space a polyhedron with 12 faces, 44 vertices, and 66 edges with this property, but it has not been accomplished. It would have irregular hendecagonal faces and genus 6.

## Gallery[edit | edit source]

The fundamental domain of the Szilassi polyhedron. Edges and vertices with the same label are identified.

## External links[edit | edit source]

- Wikipedia Contributors. "Szilassi polyhedron".
- Weisstein, Eric W. "Szilassi Polyhedron" at MathWorld.
- McCooey, David. "Szilassi Polyhedron"

- McNeil, Jim. "The Szilassi and Czászár Polyhedra"