# Rectified truncated icosahedron

Rectified truncated icosahedron | |
---|---|

Rank | 3 |

Type | Near-miss |

Space | Spherical |

Notation | |

Conway notation | atI |

Elements | |

Faces | 60 isosceles triangles, 12 pentagons, 20 triambi |

Edges | 60+120 |

Vertices | 30+60 |

Vertex figures | 60 rectangles |

30 isosceles trapezoids | |

Measures (based on regular hexagons of edge length 1) | |

Edge length ratio | |

Central density | 1 |

Number of external pieces | 28 |

Related polytopes | |

Dual | Rhombic enneacontahedron |

Conjugate | None |

Abstract & topological properties | |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | H_{3}, order 120 |

Convex | Yes |

Nature | Tame |

The **rectified truncated icosahedron** is a near-miss Johnson solid. Topologically, its faces are 60 isosceles triangles, 12 pentagons and 20 triambi. It can be constructed by applying a rectification process to the truncated icosahedron.

## Variations[edit | edit source]

There are three notable variants of the rectified truncated icosahedron. The first variant is obtained by making the hexagons regular, in which the ratio between the two edges is equivalent to 1: ≈ 1:1.07013. The second variant is obtained by making all the edges have the same length, but the hexagons only have triangular symmetry, with the internal angles being approximately 115.28055° and 124.71945°. The third variant has the unique property of being canonical. If the canonical variant has a midradius of 1, then the edge lengths are approximately 0.34561 and 0.35956 and the hexagons' internal angles are approximately 118.07524° and 121.92476°.

## Net:[edit | edit source]

## External links[edit | edit source]

- McCooey, David. "Rectified Truncated Icosahedron"

- Kaplan, Craig S. Near Misses