Dodecadodecahedron

Dodecadodecahedron
Type	Uniform star polyhedron
Elements	F = 24, E = 60 V = 30 (χ = −6)
Faces by sides	12{5}+12{5/2}
Coxeter diagram
Wythoff symbol	2 | 5 5/2 2 | 5 5/3 2 | 5/2 5/4 2 | 5/3 5/4
Symmetry group	Ih, [5,3], *532
Index references	U36, C45, W73
Dual polyhedron	Medial rhombic triacontahedron
Vertex figure	5.5/2.5.5/2
Bowers acronym	Did

In

The edges of this model form 10 central hexagons, and these, projected onto a sphere, become 10 great circles. These 10, along with the great circles from projections of two other polyhedra, form the 31 great circles of the spherical icosahedron used in construction of geodesic domes.

Wythoff constructions

It has four

Net

A shape with the same exterior appearance as the dodecadodecahedron can be constructed by folding up these nets:

12 pentagrams and 20 rhombic clusters are necessary. However, this construction replaces the crossing pentagonal faces of the dodecadodecahedron with non-crossing sets of rhombi, so it does not produce the same internal structure.

Related polyhedra

Its

(having the pentagonal faces in common).

Dodecadodecahedron	Small dodecahemicosahedron
Great dodecahemicosahedron	Icosidodecahedron (convex hull)

This polyhedron can be considered a rectified great dodecahedron. It is center of a truncation sequence between a small stellated dodecahedron and great dodecahedron:

The

.

Name	Small stellated dodecahedron	Truncated small stellated dodecahedron	Dodecadodecahedron	Truncated great dodecahedron	Great dodecahedron
Coxeter-Dynkin diagram
Picture

It is topologically equivalent to a quotient space of the hyperbolic order-4 pentagonal tiling, by distorting the pentagrams back into regular pentagons. As such, it is topologically a regular polyhedron of index two:^[2][3]

Graphs of the dodecadodecahedron (left) and its dual (right) drawn in Bring's curve. The former is a quotient of the order-4 pentagonal and the latter of the order-5 square tiling. The letters (and colors) indicate, which sides of the fundamental 20-gon belong together. Faces cut by these sides are marked by colors.
24 pentagons 11 are complete, 10 are cut in half, 2 are cut in five pieces, 1 is cut in ten pieces	30 squares 20 are complete, 10 are cut in half

Medial rhombic triacontahedron

Medial rhombic triacontahedron
Type	Star polyhedron
Face
Elements	F = 30, E = 60 V = 24 (χ = −6)
Symmetry group	Ih, [5,3], *532
Index references	DU36
dual polyhedron	Dodecadodecahedron

The medial rhombic triacontahedron is the dual of the dodecadodecahedron. It has 30 intersecting rhombic faces.

Related hyperbolic tiling

It is topologically equivalent to a quotient space of the

Note that the order-5 square tiling is dual to the order-4 pentagonal tiling, and a quotient space of the order-4 pentagonal tiling is topologically equivalent to the dual of the medial rhombic triacontahedron, the dodecadodecahedron.

See also

• List of uniform polyhedra

References

• Badoureau (1881), "Mémoire sur les figures isoscèles", Journal de l'École Polytechnique, 49: 47–172
• Hess, Edmund (1878), Vier archimedeische Polyeder höherer Art, Cassel. Th. Kay,
  JFM 10.0346.03
• Pitsch (1882), "Über halbreguläre Sternpolyheder", Zeitschrift für das Realschulwesen, 7,
  JFM 14.0448.01
• MR 0730208

• Weisstein, Eric W. "Dodecadodecahedron". MathWorld.
• Weisstein, Eric W. "Medial Rhombic Triacontahedron". MathWorld.
• Uniform polyhedra and duals