Great dirhombicosidodecahedron

Great dirhombicosidodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Gidrid
Coxeter diagram	s3s5/2s3/2s5/3*aØ*c *bØ*d
Elements
Faces	40 triangles, 60 squares, 24 pentagrams
Edges	120+120
Vertices	60
Vertex figure	Irregular octagon, edge lengths 1, √2, (√5–1)/2, √2, 1, √2, (√5–1)/2, √2
Measures (edge length 1)
Circumradius	${\frac {\sqrt {2}}{2}}\approx 0.70711$
Volume	0
Dihedral angles	5/2–4: $\arccos \left({\sqrt {\frac {5-2{\sqrt {5}}}{5}}}\right)\approx 71.03929^{\circ }$
	3–4: $\arccos \left({\frac {\sqrt {3}}{3}}\right)\approx 54.73561^{\circ }$
Central density	0
Number of external pieces	3000
Level of complexity	164
Related polytopes
Army	Semi-uniform srid, edge lengths ${\sqrt {\frac {3-{\sqrt {5}}-{\sqrt {10{\sqrt {5}}-22}}}{2}}}$ (pentagons), ${\sqrt {\frac {{\sqrt {5}}-1-2{\sqrt {{\sqrt {5}}-2}}}{2}}}$ (triangles)
Regiment	Gidrid
Dual	Great dirhombicosidodecacron
Conjugate	Great dirhombicosidodecahedron
Abstract & topological properties
Flag count	960
Euler characteristic	–56
Orientable	Yes
Genus	29
Properties
Symmetry	H3, order 120
Flag orbits	8
Convex	No
Nature	Tame

The great dirhombicosidodecahedron, Miller's monster, or gidrid, is a uniform star polyhedron. It is the most complex of the ten uniform hemipolyhedra. It consists of 40 triangles, 60 snub squares, and 24 pentagrams, all of which fall into coinciding planes in pairs. Two triangles, four squares, and two pentagrams meet at each vertex.

Vertex coordinates

A great dirhombicosidodecahedron of edge length 1 has vertex coordinates given by all even permutations of:

• $\left(\pm {\sqrt {\frac {{\sqrt {5}}-1-2{\sqrt {{\sqrt {5}}-2}}}{8}}},\,\pm {\sqrt {\frac {3-{\sqrt {5}}-{\sqrt {10{\sqrt {5}}-22}}}{8}}},\,\pm {\sqrt {\frac {2+{\sqrt {2{\sqrt {5}}-2}}}{8}}}\right)$,
• $\left(0,\,\pm {\frac {\sqrt {3-{\sqrt {5}}}}{2}},\,\pm {\frac {\sqrt {{\sqrt {5}}-1}}{2}}\right)$,
• $\left(\pm {\sqrt {\frac {3-{\sqrt {5}}+{\sqrt {10{\sqrt {5}}-22}}}{8}}},\,\pm {\sqrt {\frac {2-{\sqrt {2{\sqrt {5}}-2}}}{8}}},\,\pm {\sqrt {\frac {{\sqrt {5}}-1+2{\sqrt {{\sqrt {5}}-2}}}{8}}}\right)$.

Related polyhedra

The great dirhombicosidodecahedron has the same edges as two uniform polyhedron compounds: the disnub icosahedron, the compound of 20 octahedra, and the snub pseudosnub rhombicosahedron, the compound of 20 tetrahemihexahedra. It also has the same vertices of the great snub dodecicosidodecahedron, which uses a subset of its edges.

The degenerate great disnub dirhombidodecahedron can be constructed by blending this polyhedron with the disnub icosahedron.

External links

• Bowers, Jonathan. "Polyhedron Category 6: Snubs" (#75).

• Klitzing, Richard. "gidrid".
• Wikipedia contributors. "Great dirhombicosidodecahedron".
• McCooey, David. "Great Dirhombicosidodecahedron"