# Great icosahedron

Great icosahedron | |
---|---|

Rank | 3 |

Type | Regular |

Notation | |

Bowers style acronym | Gike |

Coxeter diagram | o5/2o3x () |

Schläfli symbol | |

Elements | |

Faces | 20 triangles |

Edges | 30 |

Vertices | 12 |

Vertex figure | Pentagram, edge length 1 |

Petrie polygons | 6 skew decagrams |

Holes | 12 pentagrams |

Measures (edge length 1) | |

Circumradius | |

Edge radius | |

Inradius | |

Volume | |

Dihedral angle | |

Central density | 7 |

Number of external pieces | 180 |

Level of complexity | 9 |

Related polytopes | |

Army | Ike, edge length |

Regiment | Sissid |

Dual | Great stellated dodecahedron |

Petrie dual | Petrial great icosahedron |

φ 2 | Small stellated dodecahedron |

Conjugate | Icosahedron |

Convex core | Icosahedron |

Abstract & topological properties | |

Flag count | 120 |

Euler characteristic | 2 |

Schläfli type | {3,5} |

Orientable | Yes |

Genus | 0 |

Skeleton | Icosahedral graph |

Properties | |

Symmetry | H_{3}, order 120 |

Flag orbits | 1 |

Convex | No |

Nature | Tame |

The **great icosahedron**, or **gike**, is one of the four Kepler–Poinsot solids. It has 20 triangles as faces, joining 5 to a vertex in a pentagrammic fashion.

It has the same edges as the small stellated dodecahedron, and the same vertices as the convex icosahedron. It is also one of the stellations of the icosahedron, and the only Kepler-Poinsot solid to be a stellation of the icosahedron as opposed to the dodecahedron.

Great icosahedra appear as cells in only one of the regular star polychora, namely the great faceted hexacosichoron.

## Vertex coordinates[edit | edit source]

Its vertices are the same as those of the small stellated dodecahedron, its regiment colonel.

## Related polytopes[edit | edit source]

The great icosahedron can be constructed by joining pentagrammic pyramids to the bases of a pentagrammic retroprism, conjugate to the icosahedron's view as a pentagonal antiprism augmented with pentagonal pyramids.

### Alternative realizations[edit | edit source]

The great icosahedron is conjugate to the icosahedron. Thus they are both faithful symmetric realizations of the same abstract regular polytope, {3,5}. There are in total 6 faithful symmetric realizations of the underlying abstract polytope. The icosahedron and the great icosahedron are the only pure faithfully symmetric realizations, the others are the results of blending those two along with the hemiicosahedron.

Dimension | Components | Name |
---|---|---|

3 | Icosahedron | Icosahedron |

3 | Great icosahedron | Great icosahedron |

6 | Skew icosahedron | |

8 | ||

8 | ||

11 |

There are also realizations that are faithful but not symmetric. The particular case of 3-dimensional realizations with regular faces are called isomorphs. They have been investigated by Jim McNeill^{[1]} and others. For example, one of the pyramids of the icosahedron can be inverted, producing an irregular polyhedron that is concave but with no intersections.

### Compounds[edit | edit source]

Two uniform polyhedron compounds are composed of great icosahedra:

### Variations[edit | edit source]

The great icosahedron can also be considered to be a kind of retrosnub tetrahedron, by analogy with the snub cube and snub dodecahedron. It is the result of alternating the vertices of a degenerate uniform polyhedron with 8 degenerate hexagrams and 6 doubled-up squares and then adjusting edge lengths to be equal. It can be represented as s3/2s3/2s or s3/2s4o, with chiral tetrahedral and pyritohedral symmetry respectively, the conjugate of the icosahedron being viewed as a snub tetrahedron.

### In vertex figures[edit | edit source]

The great icosahedron appears as a vertex figure of two Schläfli–Hess polychora.

Name | Picture | Schläfli symbol | Edge length |
---|---|---|---|

Grand hexacosichoron | {3,3,5/2} | ||

Grand hecatonicosachoron | {5,3,5/2} |

## External links[edit | edit source]

- Bowers, Jonathan. "Polyhedron Category 1: Regulars" (#8).

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

- Klitzing, Richard. "Gike".
- Nan Ma. "Great icosahedron {3, 5/2}".

- Wikipedia contributors. "Great icosahedron".
- McCooey, David. "Great Icosahedron"
- Hartley, Michael. "{3,5}*120".

- ↑ Jim McNeill. Isomorphs of the icosahedron