# Truncated icosahedron

Truncated icosahedron | |
---|---|

Rank | 3 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Ti |

Coxeter diagram | o5x3x () |

Elements | |

Faces | 12 pentagons, 20 hexagons |

Edges | 30+60 |

Vertices | 60 |

Vertex figure | Isosceles triangle, edge lengths (1+√5)/2, √3, √3 |

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 6–5: |

6–6: | |

Central density | 1 |

Number of pieces | 32 |

Level of complexity | 3 |

Related polytopes | |

Army | Ti |

Regiment | Ti |

Dual | Pentakis dodecahedron |

Conjugate | Truncated great icosahedron |

Abstract properties | |

Flag count | 360 |

Euler characteristic | 2 |

Topological properties | |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | H_{3}, order 120 |

Convex | Yes |

Nature | Tame |

The **truncated icosahedron**, or **ti**, is one of the 13 Archimedean solids. It consists of 12 pentagons and 20 hexagons. Each vertex joins one pentagon and two hexagons. As the name suggests, it can be obtained by the truncation of the icosahedron. It is the shape of a soccer ball.

## Vertex coordinates[edit | edit source]

A truncated icosahedron of edge length 1 has vertex coordinates given by all even permutations and all changes of sign of:

## Representations[edit | edit source]

A truncated icosahedron has the following Coxeter diagrams:

- o5x3x (full symmetry)
- xuxuxfoo5oofxuxux&#xt (H2 axial, pentagon-first)
- xuAxBfVoVofx3xfoVoVfBxAux&#xt (A2 axial, hexagon-first)

## Semi-uniform variant[edit | edit source]

The truncated icosahedron has a semi-uniform variant of the form o5y3x that maintains its full symmetry. This variant has 12 pentagons of size y and 20 ditrigons as faces.

With edges of length a (between two ditrigons) and b (between a ditrigon and a pentagon), its circumradius is given by and its volume is given by .

## Related polyhedra[edit | edit source]

Name | OBSA | Schläfli symbol | CD diagram | Picture |
---|---|---|---|---|

Dodecahedron | doe | {5,3} | x5o3o | |

Truncated dodecahedron | tid | t{5,3} | x5x3o | |

Icosidodecahedron | id | r{5,3} | o5x3o | |

Truncated icosahedron | ti | t{3,5} | o5x3x | |

Icosahedron | ike | {3,5} | o5o3x | |

Small rhombicosidodecahedron | srid | rr{5,3} | x5o3x | |

Great rhombicosidodecahedron | grid | tr{5,3} | x5x3x | |

Snub dodecahedron | snid | sr{5,3} | s5s3s |

## External links[edit | edit source]

- Bowers, Jonathan. "Polyhedron Category 2: Truncates" (#14).

- Klitzing, Richard. "ti".

- Quickfur. "The Truncated Icosahedron".

- Wikipedia Contributors. "Truncated icosahedron".
- McCooey, David. "Truncated Icosahedron"

- Hi.gher.Space Wiki Contributors. "Icosahedral truncate".