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Inverted snub dodecadodecahedron

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Inverted 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 | 5/3 2 5
Symmetry group I, [5,3]+, 532
Index references U60, C76, W114
Dual polyhedron Medial inverted pentagonal hexecontahedron
Vertex figure
3.3.5.3.5/3
Bowers acronym Isdid
3D model of an inverted snub dodecadodecahedron

In geometry, the inverted snub dodecadodecahedron (or vertisnub dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U60.[1] It is given a Schläfli symbol sr{5/3,5}.

Cartesian coordinates

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Let be the largest 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 Snub dodecadodecahedron

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Medial inverted pentagonal hexecontahedron

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Medial inverted pentagonal hexecontahedron
Type Star polyhedron
Face
Elements F = 60, E = 150
V = 84 (χ = −6)
Symmetry group I, [5,3]+, 532
Index references DU60
dual polyhedron Inverted snub dodecadodecahedron
3D model of a medial inverted pentagonal hexecontahedron

The medial inverted pentagonal hexecontahedron (or midly petaloid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform inverted snub dodecadodecahedron. Its faces are irregular nonconvex pentagons, with one very acute angle.

Proportions

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Denote the golden ratio by , and let be the largest (least negative) real zero of the polynomial . Then each face has three equal angles of , one of and one of . Each face has one medium length edge, two short and two long ones. If the medium length is , then the short edges have length and the long edges have length The dihedral angle equals . The other real zero of the polynomial plays a similar role for the medial pentagonal hexecontahedron.

See also

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References

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  • Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208 p. 124
  1. ^ Roman, Maeder. "60: inverted snub dodecadodecahedron". MathConsult.
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