Great dirhombicosidodecahedron

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Great dirhombicosidodecahedron

Type	Uniform polyhedron
Elements	F=124, E=240, V=60 (χ=-56)
Faces by sides	40{3}+60{4}+24{5/2}
Vertex configuration	(4.5/3.4.3.4.5/2.4.3/2)/2
Wythoff symbol	\|3/2 5/3 3 5/2
Symmetry group	I_h
Index references	U₇₅, C₉₂, W₁₁₉
Vertex Figure

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 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).

If the definition of a uniform polyhedron is relaxed to allow an even number of faces adjacent to an edge then this polyhedron 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 shape is also significant mathematically. At a 1972 Pasedena mathematics conference Dr. Steven McHarty, a professor of mathematics at Princeton showed using number theory that the Wythoff Conversion allows for inverse magnitutes of infinite series. This means that there are more points along any edge than are contained within the surface area of the shape, and more points in the surface area than are contained within the volume of the object.