# Great triambic icosahedron

In geometry, the **great triambic icosahedron** and **medial triambic icosahedron** (or **midly triambic icosahedron**) are visually identical dual uniform polyhedra. The exterior surface also represents the **De _{2}f_{2}** stellation of the icosahedron. The only way to differentiate these two polyhedra is to mark which intersections between edges are true vertices and which are not. In the above images, true vertices are marked by gold spheres, which can be seen in the concave Y-shaped areas.

Great triambic icosahedron |
Medial triambic icosahedron | |

Types | Dual uniform polyhedra | |

Symmetry group | I_{h} | |

Name | Great triambic icosahedron | Medial triambic icosahedron |

Index references | DU_{47}, W_{34}, 30/59 | DU_{41}, W_{34}, 30/59 |

Elements | F = 20, E = 60V = 32 (χ = -8) | F = 20, E = 60V = 24 (χ = -16) |

Isohedral faces | ||

Duals | Great ditrigonal icosidodecahedron | Ditrigonal dodecadodecahedron |

Stellation | ||

Icosahedron: W_{34} | ||

Stellation diagram |

The 12 vertices of the convex hull matches the vertex arrangement of an icosahedron.

## Great triambic icosahedron

The **great triambic icosahedron** is the dual of the great ditrigonal icosidodecahedron, U47. It has 20 inverted-hexagonal (triambus) faces, shaped like a three-bladed propeller. It has 32 vertices: 12 exterior points, and 20 hidden inside. It has 60 edges.

## Medial triambic icosahedron

The **medial triambic icosahedron** is the dual of the ditrigonal dodecadodecahedron, U41. It has 20 faces, each being simple concave isogonal hexagons or triambi. It has 24 vertices: 12 exterior points, and 12 hidden inside. It has 60 edges.

Unlike the great triambic icosahedron, the medial triambic icosahedron is topologically a regular polyhedron of index two.[1] By distorting the triambi into regular hexagons, one obtains a quotient space of the hyperbolic order-5 hexagonal tiling:

## References

- The Regular Polyhedra (of index two), David A. Richter

- Wenninger, Magnus (1974).
*Polyhedron Models*. Cambridge University Press. ISBN 0-521-09859-9. - Wenninger, Magnus (1983).
*Dual Models*. Cambridge University Press. ISBN 978-0-521-54325-5. MR 0730208. - H.S.M. Coxeter,
*Regular Polytopes*, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, 3.6 6.2*Stellating the Platonic solids*, pp.96-104

## External links

- Weisstein, Eric W. "Great triambic icosahedron".
*MathWorld*. - Weisstein, Eric W. "Medial triambic icosahedron".
*MathWorld*. - gratrix.net Uniform polyhedra and duals
- bulatov.org Medial triambic icosahedron Great triambic icosahedron

Notable stellations of the icosahedron | |||||||||

Regular | Uniform duals | Regular compounds | Regular star | Others | |||||

(Convex) icosahedron | Small triambic icosahedron | Medial triambic icosahedron | Great triambic icosahedron | Compound of five octahedra | Compound of five tetrahedra | Compound of ten tetrahedra | Great icosahedron | Excavated dodecahedron | Final stellation |
---|---|---|---|---|---|---|---|---|---|

The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry. |