Inverted snub dodecadodecahedron
Inverted snub dodecadodecahedron | |
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![]() | |
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 |
![](http://upload.wikimedia.org/wikipedia/commons/thumb/3/37/Inverted_snub_dodecadodecahedron.stl/220px-Inverted_snub_dodecadodecahedron.stl.png)
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
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
Related polyhedra
Medial inverted pentagonal hexecontahedron
Medial inverted pentagonal hexecontahedron | |
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![]() | |
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 |
![](http://upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Medial_inverted_pentagonal_hexecontahedron.stl/220px-Medial_inverted_pentagonal_hexecontahedron.stl.png)
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
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
References
- Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208 p. 124
- ^ Roman, Maeder. "60: inverted snub dodecadodecahedron". MathConsult.
External links
- Weisstein, Eric W. "Medial inverted pentagonal hexecontahedron". MathWorld.
- Weisstein, Eric W. "Inverted snub dodecadodecahedron". MathWorld.
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