Great ditrigonal icosidodecahedron

In geometry, the great ditrigonal icosidodecahedron (or great ditrigonary icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U47. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 20 vertices.[1] It has 4 Schwarz triangle equivalent constructions, for example Wythoff symbol 3 | 3 ​5⁄4 gives Coxeter diagram = . It has extended Schläfli symbol a{​5⁄2,3} or c{3,​5⁄2}, as an altered great stellated dodecahedron or converted great icosahedron.

Great ditrigonal icosidodecahedron
TypeUniform star polyhedron
ElementsF = 32, E = 60
V = 20 (χ = −8)
Faces by sides20{3}+12{5}
Wythoff symbol3/2 | 3 5
3 | 3/2 5
3 | 3 5/4
3/2 | 3/2 5/4
Symmetry groupIh, [5,3], *532
Index referencesU47, C61, W87
Dual polyhedronGreat triambic icosahedron
Vertex figure
((3.5)3)/2
Bowers acronymGidtid
3D model of a great ditrigonal icosidodecahedron

Its circumradius is ​√3⁄2 times the length of its edge,[2] a value it shares with the cube.

Its convex hull is a regular dodecahedron. It additionally shares its edge arrangement with the small ditrigonal icosidodecahedron (having the triangular faces in common), the ditrigonal dodecadodecahedron (having the pentagonal faces in common), and the regular compound of five cubes.

a{5,3} a{5/2,3} b{5,5/2}
= =

Small ditrigonal icosidodecahedron
Great ditrigonal icosidodecahedron
Ditrigonal dodecadodecahedron

Dodecahedron (convex hull)
Compound of five cubes

References
  1. Maeder, Roman. "47: great ditrigonal icosidodecahedron". MathConsult.
  2. Weisstein, Eric W (2003), CRC concise encyclopedia of mathematics, Boca Raton: Chapman & Hall/CRC, ISBN 1-58488-347-2
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