Snub icosidodecadodecahedron

Snub icosidodecadodecahedron
Type Uniform star polyhedron
Elements F = 104, E = 180
V = 60 (χ = −16)
Faces by sides (20+60){3}+12{5}+12{5/2}
Coxeter diagram
Wythoff symbol | 5/3 3 5
Symmetry group I, [5,3]+, 532
Index references U46, C58, W112
Dual polyhedron Medial hexagonal hexecontahedron
Vertex figure
3.3.3.5.3.5/3
Bowers acronym Sided

In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U46. It has 104 faces (80 triangles, 12 pentagons, and 12 pentagrams), 180 edges, and 60 vertices.[1] As the name indicates, it belongs to the family of snub polyhedra.

3D model of a snub icosidodecadodecahedron

Cartesian coordinates

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Let   be the real zero of the polynomial  . The number   is known as the plastic ratio. Denote by   the golden ratio. Let the point   be given by

 .

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 snub icosidodecadodecahedron. The edge length equals  , the circumradius equals  , and the midradius equals  .

For a snub icosidodecadodecahedron whose edge length is 1, the circumradius is

 

Its midradius is

 
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Medial hexagonal hexecontahedron

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Medial hexagonal hexecontahedron
 
Type Star polyhedron
Face  
Elements F = 60, E = 180
V = 104 (χ = −16)
Symmetry group I, [5,3]+, 532
Index references DU46
dual polyhedron Snub icosidodecadodecahedron
 
3D model of a medial hexagonal hexecontahedron

The medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron.

See also

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References

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  1. ^ Maeder, Roman. "46: snub icosidodecadodecahedron". MathConsult.
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