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
Type	Uniform star polyhedron
Elements	F = 24, E = 60 V = 20 (χ = −16)
Faces by sides	12{5}+12{5/2}
Wythoff symbol	3 | 5/3 5 3/2 | 5 5/2 3/2 | 5/3 5/4 3 | 5/2 5/4
Symmetry group	Ih, [5,3], *532
Index references	U41, C53, W80
Dual polyhedron	Medial triambic icosahedron
Vertex figure	(5.5/3)^3
Bowers acronym	Ditdid

3D model of a ditrigonal dodecadodecahedron

Related polyhedra

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]

See also

• List of uniform polyhedra

References

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

External links

• Weisstein, Eric W. "Ditrigonal dodecadodecahedron". MathWorld.