Great dodecicosahedron

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

Great dodecicosahedron
TypeUniform star polyhedron
ElementsF = 32, E = 120
V = 60 (χ = 28)
Faces by sides20{6}+12{10/3}
Wythoff symbol3 5/3 (3/2 5/2) |
Symmetry groupIh, [5,3], *532
Index referencesU63, C79, W101
Dual polyhedronGreat dodecicosacron
Vertex figure
6.10/3.6/5.10/7
Bowers acronymGiddy

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.

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

Great icosicosidodecahedron

Great ditrigonal dodecicosidodecahedron

Great dodecicosahedron

Filling

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]


Traditional filling

"Neo filling"

See also

References

  • Weisstein, Eric W. "Great dodecicosahedron". MathWorld.
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