Great snub dodecicosidodecahedron

Great snub dodecicosidodecahedron
Type Uniform star polyhedron
Elements F = 104, E = 180
V = 60 (χ = −16)
Faces by sides (20+60){3}+(12+12){5/2}
Coxeter diagram
Wythoff symbol | 5/3 5/2 3
Symmetry group I, [5,3]+, 532
Index references U64, C80, W115
Dual polyhedron Great hexagonal hexecontahedron
Vertex figure
3.3.3.5/2.3.5/3
Bowers acronym Gisdid

In geometry, the great snub dodecicosidodecahedron (or great snub dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U64. It has 104 faces (80 triangles and 24 pentagrams), 180 edges, and 60 vertices.[1] It has Coxeter diagram . It has the unusual feature that its 24 pentagram faces occur in 12 coplanar pairs.

3D model of a great snub dodecicosidodecahedron

Cartesian coordinates

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Let the point   be given by

 ,

where   is the golden ratio. Let the matrix   be given by

 .

  is the rotation around the axis   by an angle of  , counterclockwise. Let the linear transformations   be the transformations which send a point   to the even permutations of   with an even number of minus signs. The transformations   constitute the group of rotational symmetries of a regular tetrahedron. The transformations    ,   constitute the group of rotational symmetries of a regular icosahedron. Then the 60 points   are the vertices of a great snub dodecicosidodecahedron. The edge length equals  , the circumradius equals  , and the midradius equals  .

For a great snub dodecicosidodecahedron whose edge length is 1, the circumradius is

 .

Its midradius is

 .
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It shares its vertices and edges, as well as 20 of its triangular faces and all its pentagrammic faces, with the great dirhombicosidodecahedron, (although the latter has 60 edges not contained in the great snub dodecicosidodecahedron). It shares its other 60 triangular faces (and its pentagrammic faces again) with the great disnub dirhombidodecahedron.

The edges and triangular faces also occur in the compound of twenty octahedra. In addition, 20 of the triangular faces occur in one enantiomer of the compound of twenty tetrahemihexahedra, and the other 60 triangular faces occur in the other enantiomer.

 
Convex hull
 
Great snub dodecicosidodecahedron
 
Great dirhombicosidodecahedron
 
Great disnub dirhombidodecahedron
 
Compound of twenty octahedra
 
Compound of twenty tetrahemihexahedra
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Traditional filling
 
Modulo-2 filling

See also

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References

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  1. ^ Maeder, Roman. "64: great snub dodecicosidodecahedron". MathConsult.
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