Ditrigonal dodecadodecahedron

In geometry, the ditrigonal dodecadodecahedron (or ditrigonary dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U41. It has 24 faces (12 pentagons and 12 pentagrams), 60 edges, and 20 vertices.[1] It has extended Schläfli symbol b{5,​5⁄2}, as a blended great dodecahedron, and Coxeter diagram . It has 4 Schwarz triangle equivalent constructions, for example Wythoff symbol 3 | ​5⁄3 5, and Coxeter diagram .

Ditrigonal dodecadodecahedron
TypeUniform star polyhedron
ElementsF = 24, E = 60
V = 20 (χ = −16)
Faces by sides12{5}+12{5/2}
Wythoff symbol3 | 5/3 5
3/2 | 5 5/2
3/2 | 5/3 5/4
3 | 5/2 5/4
Symmetry groupIh, [5,3], *532
Index referencesU41, C53, W80
Dual polyhedronMedial triambic icosahedron
Vertex figure
(5.5/3)3
Bowers acronymDitdid
3D model of a ditrigonal dodecadodecahedron

Its convex hull is a regular dodecahedron. It additionally shares its edge arrangement with the small ditrigonal icosidodecahedron (having the pentagrammic faces in common), the great ditrigonal icosidodecahedron (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

Furthermore, it may be viewed as a facetted dodecahedron: the pentagonal faces may be inscribed within the dodecahedron's pentagons. Its dual, the medial triambic icosahedron, is a stellation of the icosahedron.

It is topologically equivalent to a quotient space of the hyperbolic order-6 pentagonal tiling, by distorting the pentagrams back into regular pentagons. As such, it is a regular polyhedron of index two:[2]

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See also

References
  1. Maeder, Roman. "41: ditrigonal dodecadodecahedron". MathConsult.
  2. The Regular Polyhedra (of index two), David A. Richter
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