Snub icosidodecadodecahedron
Snub icosidodecadodecahedron | |
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![]() | |
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 |
![](http://upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Snub_icosidodecadodecahedron.stl/220px-Snub_icosidodecadodecahedron.stl.png)
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.
Cartesian coordinates
[edit]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
Related polyhedra
[edit]Medial hexagonal hexecontahedron
[edit]Medial hexagonal hexecontahedron | |
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![]() | |
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 |
![](http://upload.wikimedia.org/wikipedia/commons/thumb/3/35/Medial_hexagonal_hexecontahedron.stl/220px-Medial_hexagonal_hexecontahedron.stl.png)
The medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron.
See also
[edit]References
[edit]- ^ Maeder, Roman. "46: snub icosidodecadodecahedron". MathConsult.
- Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208
External links
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