Jump to content

Ditrigonal dodecadodecahedron

From Wikipedia, the free encyclopedia
Ditrigonal dodecadodecahedron
Type Uniform star polyhedron
Elements F = 24, E = 60
V = 20 (χ = −16)
Faces by sides 12{5}+12{5/2}
Coxeter diagram
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

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{5,52}, as a blended great dodecahedron, and Coxeter diagram . It has 4 Schwarz triangle equivalent constructions, for example Wythoff symbol 3 | 53 5, and Coxeter diagram .

[edit]

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{52,3} b{5,52}
= = =

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 pentagrammic faces are inscribed in 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

[edit]

References

[edit]
  1. ^ Maeder, Roman. "41: ditrigonal dodecadodecahedron". MathConsult. Archived from the original on 2015-09-21.
  2. ^ The Regular Polyhedra (of index two) Archived 2016-03-04 at the Wayback Machine, David A. Richter
[edit]