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}
Coxeter diagram
Wythoff symbol	| 3/2 5/3 3 5/2
Symmetry group	Ih, [5,3], *532
Index references	U75, C92, W119
Dual polyhedron	Great dirhombicosidodecacron
Vertex figure	4.5/3.4.3.4.5/2.4.3/2
Bowers acronym	Gidrid

In geometry, the great dirhombicosidodecahedron (or great snub disicosidisdodecahedron) is a nonconvex uniform polyhedron, indexed last as U₇₅. It has 124 faces (40 triangles, 60 squares, and 24 pentagrams), 240 edges, and 60 vertices.^[1]

This is the only non-degenerate 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. Another unusual feature is that the faces all occur in coplanar pairs.

This is also the only uniform polyhedron that cannot be made by the Wythoff construction from a spherical triangle. It has a special Wythoff symbol | 3⁄2 5⁄3 3 5⁄2, relating it to a spherical quadrilateral. This symbol suggests that it is a sort of snub polyhedron, except that instead of the non-snub faces being surrounded by snub triangles as in most snub polyhedra, they are surrounded by snub squares.

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).

Related polyhedra

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

This polyhedron is related to the nonconvex great rhombicosidodecahedron (quasirhombicosidodecahedron) by a branched cover: there is a function from the great dirhombicosidodecahedron to the quasirhombicosidodecahedron that is 2-to-1 everywhere, except for the vertices.^[2]

Cartesian coordinates

Cartesian coordinates for the vertices of a great dirhombicosidodecahedron are all the even permutations of

$${\displaystyle {\begin{array}{crrrc}{\Bigl (}&0\ ,&\pm {\frac {2}{\varphi ))\ ,&\pm {\frac {2}{\sqrt {\varphi ))}&{\Bigr )},\\{\Bigl (}&\pm \left[-1+{\frac {1}{\varphi ^{3/2))}\right],&\pm \left[{\frac {1}{\varphi ^{2))}-{\frac {1}{\sqrt {\varphi ))}\right],&\pm \left[{\frac {1}{\varphi ))+{\sqrt {\varphi ))\right]&{\Bigr )},\\{\Bigl (}&\pm \left[-{\frac {1}{\varphi ))+{\sqrt {\varphi ))\right],&\pm \left[-1-{\frac {1}{\varphi ^{3/2))}\right],&\pm \left[{\frac {1}{\varphi ^{2))}+{\frac {1}{\sqrt {\varphi ))}\right]&{\Bigr )},\end{array))}$$

where ${\displaystyle \varphi ={\tfrac {1+{\sqrt {5))}{2))}$ is the golden ratio. These vertices result in an edge length of ${\displaystyle 2{\sqrt {2)).}$

Gallery

Traditional filling

Modulo-2 filling

Interior view, modulo-2 filling