Great dodecicosahedron

Great dodecicosahedron
Type	Uniform star polyhedron
Elements	F = 32, E = 120 V = 60 (χ = −28)
Faces by sides	20{6}+12{10/3}
Coxeter diagram	(with extra double-covered triangles) (with extra double-covered pentagons)
Wythoff symbol	3 5/3 (3/2 5/2) |
Symmetry group	Ih, [5,3], *532
Index references	U63, C79, W101
Dual polyhedron	Great dodecicosacron
Vertex figure	6.10/3.6/5.10/7
Bowers acronym	Giddy

In geometry, the great dodecicosahedron (or great dodekicosahedron) is a nonconvex uniform polyhedron, indexed as U₆₃. It has 32 faces (20 hexagons and 12 decagrams), 120 edges, and 60 vertices.^[1] Its vertex figure is a crossed quadrilateral.

It has a composite Wythoff symbol, 3 5⁄3 (3⁄2 5⁄2) |, requiring two different Schwarz triangles to generate it: (3 5⁄3 3⁄2) and (3 5⁄3 5⁄2). (3 5⁄3 3⁄2 | represents the great dodecicosahedron with an extra 12 {10⁄2} pentagons, and 3 5⁄3 5⁄2 | represents it with an extra 20 {6⁄2} triangles.)^[2]

Its vertex figure 6.10⁄3.6⁄5.10⁄7 is also ambiguous, having two clockwise and two counterclockwise faces around each vertex.

It shares its vertex arrangement with the truncated dodecahedron. It additionally shares its edge arrangement with the great icosicosidodecahedron (having the hexagonal faces in common) and the great ditrigonal dodecicosidodecahedron (having the decagrammic faces in common).

Truncated dodecahedron

Great icosicosidodecahedron

Great ditrigonal dodecicosidodecahedron

Great dodecicosahedron

Traditional filling

Modulo-2 filling

• List of uniform polyhedra

• Weisstein, Eric W. "Great dodecicosahedron". MathWorld.

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