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Great inverted snub icosidodecahedron

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Great inverted snub icosidodecahedron
Type Uniform star polyhedron
Elements F = 92, E = 150
V = 60 (χ = 2)
Faces by sides (20+60){3}+12{5/2}
Coxeter diagram
Wythoff symbol | 5/3 2 3
Symmetry group I, [5,3]+, 532
Index references U69, C73, W116
Dual polyhedron Great inverted pentagonal hexecontahedron
Vertex figure
34.5/3
Bowers acronym Gisid
3D model of a great inverted snub icosidodecahedron

In geometry, the great inverted snub icosidodecahedron (or great vertisnub icosidodecahedron) is a uniform star polyhedron, indexed as U69. It is given a Schläfli symbol sr{53,3}, and Coxeter-Dynkin diagram . In the book Polyhedron Models by Magnus Wenninger, the polyhedron is misnamed great snub icosidodecahedron, and vice versa.

Cartesian coordinates

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Let be the largest (least negative) negative zero of the polynomial , where is 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 great snub icosahedron. 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 four positive real roots of the sextic in R2, are the circumradii of the snub dodecahedron (U29), great snub icosidodecahedron (U57), great inverted snub icosidodecahedron (U69), and great retrosnub icosidodecahedron (U74).

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Great inverted pentagonal hexecontahedron

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Great inverted pentagonal hexecontahedron
Type Star polyhedron
Face
Elements F = 60, E = 150
V = 92 (χ = 2)
Symmetry group I, [5,3]+, 532
Index references DU69
dual polyhedron Great inverted snub icosidodecahedron
3D model of a great inverted pentagonal hexecontahedron

The great inverted pentagonal hexecontahedron (or petaloidal trisicosahedron) is a nonconvex isohedral polyhedron. It is composed of 60 concave pentagonal faces, 150 edges and 92 vertices.

It is the dual of the uniform great inverted snub icosidodecahedron.

Proportions

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Denote the golden ratio by . Let be the smallest positive zero of the polynomial . Then each pentagonal face has four equal angles of and one angle of . Each face has three long and two short edges. The ratio between the lengths of the long and the short edges is given by

.

The dihedral angle equals . Part of each face lies inside the solid, hence is invisible in solid models. The other two zeroes of the polynomial play a similar role in the description of the great pentagonal hexecontahedron and the great pentagrammic 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. 126
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