Small snub icosicosidodecahedron
Small snub icosicosidodecahedron | |
---|---|
![]() | |
Type | Uniform star polyhedron |
Elements | F = 112, E = 180 V = 60 (χ = −8) |
Faces by sides | (40+60){3}+12{5/2} |
Coxeter diagram | ![]() ![]() ![]() ![]() |
Wythoff symbol | | 5/2 3 3 |
Symmetry group | Ih, [5,3], *532 |
Index references | U32, C41, W110 |
Dual polyhedron | Small hexagonal hexecontahedron |
Vertex figure | ![]() 35.5/2 |
Bowers acronym | Seside |
![](http://upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Small_snub_icosicosidododecahedron.stl/220px-Small_snub_icosicosidododecahedron.stl.png)
In geometry, the small snub icosicosidodecahedron or snub disicosidodecahedron is a uniform star polyhedron, indexed as U32. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices. Its stellation core is a truncated pentakis dodecahedron. It also called a holosnub icosahedron, ß{3,5}.
The 40 non-snub triangular faces form 20 coplanar pairs, forming star hexagons that are not quite regular. Unlike most snub polyhedra, it has reflection symmetries.
Convex hull
Its convex hull is a nonuniform truncated icosahedron.
![]() Truncated icosahedron (regular faces) | ![]() Convex hull (isogonal hexagons) | ![]() Small snub icosicosidodecahedron |
Cartesian coordinates
Let be largest (least 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 small snub icosicosidodecahedron. The edge length equals , the circumradius equals , and the midradius equals .
For a small snub icosicosidodecahedron whose edge length is 1, the circumradius is
Its midradius is
The other zero of plays a similar role in the description of the small retrosnub icosicosidodecahedron.
See also
External links
- Weisstein, Eric W. "Small snub icosicosidodecahedron". MathWorld.
- Klitzing, Richard. "3D star small snub icosicosidodecahedron".
- v
- t
- e