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 |
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{5⁄2,5}, as a snub great dodecahedron.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/3/37/Snub_dodecadodecahedron.stl/220px-Snub_dodecadodecahedron.stl.png)
Cartesian coordinates
editLet 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
Related polyhedra
editMedial pentagonal hexecontahedron
editMedial 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 |
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
editReferences
edit- ^ Maeder, Roman. "40: snub dodecadodecahedron". MathConsult.
- Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208
External links
edit- Weisstein, Eric W. "Medial pentagonal hexecontahedron". MathWorld.
- Weisstein, Eric W. "Snub dodecadodecahedron". MathWorld.