| Great dirhombicosidodecahedron | |
|---|---|
| Type | Uniform star polyhedron |
| Elements | F = 124, E = 240 V = 60 (χ = −56) |
| Faces by sides | 40{3}+60{4}+24{5/2} |
| Wythoff symbol | |3/2 5/3 3 5/2 |
| Symmetry group | Ih, [5,3], *532 |
| Index references | U75, C92, W119 |
| Bowers acronym | Gidrid |
4.5/3.4.3.4. 5/2.4.3/2 (Vertex figure) |
Great dirhombicosidodecacron (dual polyhedron) |
In geometry, the great dirhombicosidodecahedron is a nonconvex uniform polyhedron, indexed last as U75.
This is the only uniform polyhedron with more than six faces meeting at a vertex. Each vertex has 4 squares which pass through the vertex central axis (and thus through the centre of the figure), alternating with two triangles and two pentagrams.
This is also the only uniform polyhedron that cannot be made by Wythoff construction. It has a special Wythoff symbol | 3/2 5/3 3 5/2.
It has been nicknamed "Miller's monster" (after J. C. P. Miller, who with H. S. M. Coxeter and M. S. Longuet-Higgins enumerated the uniform polyhedra in 1954).
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If the definition of a uniform polyhedron is relaxed to allow any even number of faces adjacent to an edge, then this definition gives rise to one further polyhedron: the great disnub dirhombidodecahedron which has the same vertices and edges but with a different arrangement of triangular faces.
The vertices and edges are also shared with the uniform compounds of 20 octahedra or 20 tetrahemihexahedra. 180 of the 240 edges are shared with the great snub dodecicosidodecahedron.
Convex hull |
Great snub dodecicosidodecahedron |
Great dirhombicosidodecahedron |
Great disnub dirhombidodecahedron |
Compound of twenty octahedra |
Compound of twenty tetrahemihexahedra |
Cartesian coordinates for the vertices of a great dirhombicosidodecahedron are all the even permutations of
where τ = (1+√5)/2 is the golden ratio (sometimes written φ). These vertices result in an edge length of 2√2.