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In geometry, the great dirhombicosidodecahedron is a nonconvex uniform polyhedron, indexed last as U75.
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This is the only uniform polyhedron with more than six faces meeting at a vertex (though there are some "degenerate uniform polyhedra" with 10 faces at each 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.
Cartesian coordinates
Cartesian coordinates for the vertices of a great dirhombicosidodecahedron are all the even permutations of
(0, ±2/τ, ±2/√τ) (±(−1+1/√τ3), ±(1/τ2−1/√τ), ±(1/τ+√τ)) (±(−1/τ+√τ), ±(−1−1/√τ3, ±(1/τ2+1/√τ))where τ = (1+√5)/2 is the golden ratio (sometimes written φ). These vertices result in an edge length of 2√2.
Filling
There is some controversy on how to colour the faces of this polyhedron. Although the common way to fill in a polygon is to just colour its whole interior, this can result in some filled regions hanging as membranes over empty space. Hence, the "neo filling" is sometimes used instead as a more accurate filling. In the neo filling, orientable polyhedra are filled traditionally, but non-orientable polyhedra have their faces filled with the modulo-2 method (only odd-density regions are filled in). In addition, overlapping regions of coplanar faces can cancel each other out. Usage of the "neo filling" makes the great dirhombicosidodecahedron a hollow polyhedron.