# Toroidal icosahedron faceting

Toroidal icosahedron faceting | |
---|---|

Rank | 3 |

Type | Toroid, orbiform |

Notation | |

Bowers style acronym | Toif |

Elements | |

Faces | 6 triangles, 6 pentagons |

Edges | 6+6+12 |

Vertices | 6+6 |

Vertex figures | 6 nonconvex pentagons, edge lengths 1, (1+√5)/2, (1+√5)/2, 1, (1+√5)/2 |

6 isosceles triangles, edge lengths 1, (1+√5)/2, (1+√5)/2 | |

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 3-5: |

5-5: | |

Central density | 0 |

Related polytopes | |

Army | Ike |

Conjugate | Toroidal great icosahedron faceting |

Convex hull | Icosahedron |

Convex core | Gyroelongated triangular bipyramid |

Abstract & topological properties | |

Flag count | 96 |

Euler characteristic | 0 |

Orientable | Yes |

Genus | 1 |

Properties | |

Symmetry | (G_{2}×A_{1})/2, order 12 |

Flag orbits | 8 |

Convex | No |

The **toroidal icosahedron faceting** or **toif**, also called the **crazy tube**, is an orbiform toroid. As its name suggests, it is a faceting of the icosahedron. Its faces are 6 pentagons and 6 triangles.

It has the fewest faces and vertices of any known regular faced toroid, with 12 of each. However is not a Stewart toroid as it self-intersects. The record for the fewest faces of a Stewart toroid is 21, with the tunnelled elongated triangular cupola and the record for the fewest vertices is 15, with the tortuous tunnel.

## Vertex coordinates[edit | edit source]

Its vertex coordinates are the same as those of the icosahedron.

## Gallery[edit | edit source]

## Related polytopes[edit | edit source]

Both the toroidal icosahedron faceting and the trigonic great dodecahedron faceting have 6 pentagonal and 6 triangular faces and they share a convex hull (the icosahedron). The two can be superimposed so that their triangular faces are exactly incident in which case their pentagonal faces are complementary, forming the full set of twelve pentagons of the great dodecahedron.

The toroidal icosahedron faceting appears as the cells of the small toroidal trigonal swirlprism, a noble scaliform polychoron.

## External links[edit | edit source]

- Bowers, Jonathan. "Batch 2: Ike and Sissid Facetings" (#10 under ike).

- McNeil, Jim. "Simple Stewart toroids".
- Klitzing, Richard. "ike-facetings".