Dodecadodecahedron

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Dodecadodecahedron
Dodecadodecahedron
Type Uniform star polyhedron
Elements F = 24, E = 60
V = 30 (χ = −6)
Faces by sides 12{5}+12{5/2}
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 Dodecadodecahedron
5.5/2.5.5/2
Bowers acronym Did

In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36.

Wythoff constructions[edit]

It has four Wythoff constructions between four Schwarz triangle families: 2 | 5 5/2, 2 | 5 5/3, 2 | 5/2 5/4, 2 | 5/3 5/4, but represent identical results. Similarly it can be given four extended Schläfli symbols: t1{5/2,5}, t1{5/3,5}, t1{5/2,5/4}, and t1{5/3,5/4}. And it can also be given four Coxeter-Dynkin diagrams: CDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.pngCDel 5.pngCDel node.png, CDel node.pngCDel 5.pngCDel rat.pngCDel d3.pngCDel node 1.pngCDel 5.pngCDel node.png, CDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.pngCDel 5.pngCDel rat.pngCDel d4.pngCDel node.png, and CDel node.pngCDel 5.pngCDel rat.pngCDel d3.pngCDel node 1.pngCDel 5.pngCDel rat.pngCDel d4.pngCDel node.png.

Net[edit]

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

Dodecadodecahedron net.png

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

Related polyhedra[edit]

Its convex hull is the icosidodecahedron. It also shares its edge arrangement with the small dodecahemicosahedron (having the pentagrammic faces in common), and with the great dodecahemicosahedron (having the pentagonal faces in common).

Dodecadodecahedron.png
Dodecadodecahedron
Small dodecahemicosahedron.png
Small dodecahemicosahedron
Great dodecahemicosahedron.png
Great dodecahemicosahedron
Icosidodecahedron.png
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 truncated small stellated dodecahedron looks like a dodecahedron on the surface, but it has 24 faces: 12 pentagons from the truncated vertices and 12 overlapping as (truncated pentagrams). The truncation of the dodecadodecahedron itself is not uniform and attempting to make it uniform results in a degenerate polyhedron (that looks like a small rhombidodecahedron with {10/2} polygons filling up the dodecahedral set of holes), but it has a uniform quasitruncation, the truncated dodecadodecahedron.

Name Small stellated dodecahedron Truncated small stellated dodecahedron Dodecadodecahedron Truncated
great
dodecahedron
Great
dodecahedron
Coxeter-Dynkin
diagram
CDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png
Picture Small stellated dodecahedron.png Dodecahedron.png Dodecadodecahedron.png Great truncated dodecahedron.png Great dodecahedron.png

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:[1][2]

Uniform tiling 552-t1.png

The colours in the above image correspond to the red pentagrams and yellow pentagons of the dodecadodecahedron at the top of this article.


Medial rhombic triacontahedron[edit]

Medial rhombic triacontahedron
Dodecadodecahedron
Type Star polyhedron
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 a nonconvex isohedral polyhedron. It is the dual of the dodecadodecahedron. It has 30 intersecting rhombic faces.

It can also be called the small stellated triacontahedron.

Stellation[edit]

The Medial rhombic triacontahedron is a stellation of the rhombic triacontahedron. The convex hull of the dodecadodecahedron is an icosidodecahedron.

Related hyperbolic tiling[edit]

It is topologically equivalent to a quotient space of the hyperbolic order-5 square tiling, by distorting the rhombi into squares. As such, it is topologically a regular polyhedron of index two:[3]

Uniform tiling 45-t0.png

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[edit]

References[edit]

External links[edit]