Great dodecicosahedron

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Great dodecicosahedron
Great dodecicosahedron.png
Type Uniform star polyhedron
Elements F = 32, E = 120
V = 60 (χ = −28)
Faces by sides 20{6}+12{10/3}
Wythoff symbol 3 5/3 (3/2 5/2) |
Symmetry group Ih, [5,3], *532
Index references U63, C79, W101
Dual polyhedron Great dodecicosacron
Vertex figure Great dodecicosahedron vertfig.png
6.10/3.6/5.10/7
Bowers acronym Giddy

In geometry, the great dodecicosahedron is a nonconvex uniform polyhedron, indexed as U63. Its vertex figure is a crossed quadrilateral.

It has a composite Wythoff symbol, 3 5/3 (3/2 5/2) |, requiring two different Schwarz triangles to generate it: (3 5/3 3/2) and (3 5/3 5/2). (3 5/3 3/2 | represents the great dodecicosahedron with an extra 12 {10/2} pentagons, and 3 5/3 5/2 | represents it with an extra 20 {6/2} triangles.)

Its vertex figure 6.10/3.6/5.10/7 is also ambiguous, having two clockwise and two counterclockwise faces around each vertex.

Related polyhedra[edit]

It shares its vertex arrangement with the truncated dodecahedron. It additionally shares its edge arrangement with the great icosicosidodecahedron (having the hexagonal faces in common) and the great ditrigonal dodecicosidodecahedron (having the decagrammic faces in common).

Truncated dodecahedron.png
Truncated dodecahedron
Great icosicosidodecahedron.png
Great icosicosidodecahedron
Great ditrigonal dodecicosidodecahedron.png
Great ditrigonal dodecicosidodecahedron
Great dodecicosahedron.png
Great dodecicosahedron

Filling[edit]

There is some controversy on how to colour the faces of this polyhedron. Although the common way to fill in a polygon is to just colour its whole interior, this can result in some filled regions hanging as membranes over empty space. Hence, the "neo filling" is sometimes used instead as a more accurate filling. In the neo filling, orientable polyhedra are filled traditionally, but non-orientable polyhedra have their faces filled with the modulo-2 method (only odd-density regions are filled in).[1]

Great dodecicosahedron.png
Traditional filling
Great dodecicosahedron 2.png
"Neo filling"

See also[edit]

References[edit]

External links[edit]