Great dodecicosahedron

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.

Great dodecicosahedron
TypeUniform star polyhedron
ElementsF = 32, E = 120
V = 60 (χ = −28)
Faces by sides20{6}+12{10/3}
Coxeter diagram (with extra double-covered triangles)
(with extra double-covered pentagons)
Wythoff symbol3 5/3 (3/2 5/2) |
Symmetry groupIh, [5,3], *532
Index referencesU63, C79, W101
Dual polyhedronGreat dodecicosacron
Vertex figure
6.10/3.6/5.10/7
Bowers acronymGiddy
3D model of a great dodecicosahedron

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

See also

References
  1. Maeder, Roman. "63: great dodecicosahedron". MathConsult.
  2. Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. pp. 9–10.
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