# Tridiminished icosahedron

Tridiminished icosahedron | |
---|---|

Rank | 3 |

Type | CRF |

Notation | |

Bowers style acronym | Teddi |

Coxeter diagram | xfo3oox&#xt |

Stewart notation | Y_{5}^{-3}I_{5}J _{63} |

Elements | |

Faces | 1+1+3 triangles, 3 pentagons |

Edges | 3+3+3+6 |

Vertices | 3+3+3 |

Vertex figures | 3 isosceles trapezoids, edge length 1, 1, 1, (1+√5)/2 |

3+3 isosceles triangles, edge lengths 1, (1+√5)/2, (1+√5)/2 | |

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 3-3: |

3-5: | |

5-5: | |

Central density | 1 |

Number of external pieces | 8 |

Level of complexity | 10 |

Related polytopes | |

Army | Teddi |

Regiment | Teddi |

Dual | Tri-tridiminished icosahedron |

Conjugate | Trireplenished great icosahedron |

Abstract & topological properties | |

Flag count | 60 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | A_{2}×I, order 6 |

Convex | Yes |

Nature | Tame |

The **tridiminished icosahedron** is one of the 92 Johnson solids (J_{63}). It consists of 1+1+3 triangles and 3 pentagons. It can be constructed by removing 3 mutually non-adjacent vertices from a regular icosahedron.

Its 9 vertices fall in three parallel planes in sets of 3. The outer planes contain the extreme triangles, while the plane between them intersects with the figure in another triangle with an edge length times the edge length of the polyhedron. This observation led to a generalization known as the ursatopes, which have vertices falling in three hyperplanes of any dimension; some ursatopes are CRF as the tridiminished icosahedron is.

## Vertex coordinates[edit | edit source]

A tridiminished icosahedron of edge length 1 has the following vertices:

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

These are the vertices of an icosahedron, but with three missing.

An alternate set of coordinates can be given in a way that positions the tridiminished icosahedron within the symmetry axis:

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

## Related polyhedra[edit | edit source]

A tetrahedron can be attached to the tridiminished icosahedron at the triangular face surrounded by pentagons to form the augmented tridiminished icosahedron.

## In vertex figures[edit | edit source]

The tridiminished icosahedron is the vertex figure of the uniform snub disicositetrachoron.

## External links[edit | edit source]

- Bowers, Jonathan. "Batch 2: Ike and Sissid Facetings" (#11 under ike).

- Klitzing, Richard. "teddi".
- Quickfur. "The Tridiminished Icosahedron".

- Wikipedia contributors. "Tridiminished icosahedron".