# Toroidal blend of 8 dodecahedra

Toroidal blend of 8 dodecahedra | |
---|---|

Rank | 3 |

Type | Stewart toroid |

Elements | |

Faces | 4+4+4+8 +4+4+4+8 +4+4+8+8+8+8 pentagons |

Edges | 2+2+4+4+8+8+8+8+8+8 +2+2+4+4+8+8+8+8+8+8 +4+4+8+8+8+8+8+8+8+8+8 |

Vertices | 4+4+4+4+8+8+8 +4+4+4+4+8+8+8 +8+8+8+8+8 |

Vertex figures | 4+4+4+4+4+4+4+8+8+8+8+8+8+8 triangles, edge length (1+√5)/2 |

4+4+8+8+8 [5.5.5.5] | |

4 [5.5.5.5.5] | |

Measures (edge length 1) | |

Volume | |

Surface area | |

Central density | 0 |

Number of external pieces | 80 |

Level of complexity | 20 |

Related polytopes | |

Convex hull | Minkowski sum of dodecahedron with edge length 1 and rhombus with edge length and diagonal ratio (1+√5)/2, where the diagonals are parallel to edges of the dodecahedron |

Abstract & topological properties | |

Flag count | 800 |

Euler characteristic | 0 |

Surface | Torus |

Orientable | Yes |

Genus | 1 |

Properties | |

Symmetry | K_{3}, order 8 |

Convex | No |

Nature | Tame |

The **toroidal blend of 8 dodecahedra** is a Stewart toroid that consists of 80 pentagons. It can be obtained by outer-blending eight dodecahedra together in a rhombus-shaped loop. The four dodecahedra at the vertices of the virtual rhombus are all oriented in the same way.

## Relations[edit | edit source]

The shape of this toroid differs from the toroidal blend of 8 octahedra: the latter's virtual rhombus has a diagonal ratio of √2, while this one's has diagonal ratio (1+√5)/2.

Other versions of the toroid can be made out of copies of most of the Archimedean or Johnson solids that share the faceplanes of the dodecahedron.

Thirty copies of the toroid can be blended together, each blending pair coinciding at three dodecahedra with collinear centers, to form a toroidal blend of 92 dodecahedra with dodecahedral symmetry and an appearance like the skeleton of a rhombic triacontahedron.

## External links[edit | edit source]

- Doskey, Alex. "Chapter 7 - Exploration of (R)(A) Toroids".
- Parker, Matt. “Polygons of New York”.