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Snub dodecadodecahedron

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Snub dodecadodecahedron
Type Uniform star polyhedron
Elements F = 84, E = 150
V = 60 (χ = −6)
Faces by sides 60{3}+12{5}+12{5/2}
Coxeter diagram
Wythoff symbol | 2 5/2 5
Symmetry group I, [5,3]+, 532
Index references U40, C49, W111
Dual polyhedron Medial pentagonal hexecontahedron
Vertex figure
3.3.5/2.3.5
Bowers acronym Siddid
3D model of a snub dodecadodecahedron

In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U40. It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices.[1] It is given a Schläfli symbol sr{52,5}, as a snub great dodecahedron.

Cartesian coordinates

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Let be the smallest real zero of the polynomial . 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 dodecadodecahedron. The edge length equals , the circumradius equals , and the midradius equals .

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

Its midradius is

The other real root of P plays a similar role in the description of the Inverted snub dodecadodecahedron

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Medial pentagonal hexecontahedron

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Medial pentagonal hexecontahedron
Type Star polyhedron
Face
Elements F = 60, E = 150
V = 84 (χ = −6)
Symmetry group I, [5,3]+, 532
Index references DU40
dual polyhedron Snub dodecadodecahedron
3D model of a medial pentagonal hexecontahedron

The medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.

See also

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References

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  1. ^ Maeder, Roman. "40: snub dodecadodecahedron". MathConsult.
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