Great dirhombicosidodecahedron

In geometry, the great dirhombicosidodecahedron is a nonconvex uniform polyhedron, indexed last as. It has 124 faces, 240 edges, and 60 vertices.
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, alternating with two triangles and two pentagrams. Since all its 60 double-square facial planes pass through its center, the great dirhombicosidodecahedron classifies as a hemipolyhedron; however, the great dirhombicosidedecahedron does not form a hemi-face like the simpler hemipolyhedra that are based on the two quasiregular polyhedra. 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 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".

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 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.

Cartesian coordinates

Let the point be given by
where is the golden ratio.
Let the matrix be given by
is the rotation around the axis by an angle of, counterclockwise. Let the linear transformations
be the transformations which send a point to the even permutations of with an even number of minus signs.
The transformations constitute the group of rotational symmetries of a regular tetrahedron.
The transformations, constitute the group of rotational symmetries of a regular icosahedron.
Then the 60 points are the vertices of a great dirhombicosidodecahedron. The edge length equals, the circumradius equals, and the midradius equals.
For a great dirhombicosidodecahedron whose edge length is 1,
the circumradius is
Its midradius is