| Small retrosnub 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 | | 3/2 3/2 5/2 |
| Symmetry group | Ih, [5,3], *532 |
| Index references | U72, C91, W118 |
| Dual polyhedron | Small hexagrammic hexecontahedron |
| Vertex figure |  (35.5/3)/2 |
| Bowers acronym | Sirsid |
In geometry, the small retrosnub icosicosidodecahedron (also known as a retrosnub disicosidodecahedron, small inverted retrosnub icosicosidodecahedron, or retroholosnub icosahedron) is a nonconvex uniform polyhedron, indexed as U72. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices.[1] It is given a Schläfli symbol sr{⁵/₃,³/₂}.
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.
George Olshevsky nicknamed it the yog-sothoth (after the Cthulhu Mythos deity).[2][3]
Convex hull
Its convex hull is a nonuniform truncated dodecahedron.
 Truncated dodecahedron |
 Convex hull |
Small retrosnub icosicosidodecahedron |
Cartesian coordinates
Cartesian coordinates for the vertices of a small retrosnub icosicosidodecahedron are all the even permutations of
![{\displaystyle {\begin{array}{crrrc}{\Bigl (}&\pm {\bigl [}1-\varphi -\alpha {\bigr ]},&0\,,&\pm {\bigl [}3-\varphi \alpha {\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\varphi -1-\alpha {\bigr ]},&\pm \,2\,,&\pm {\bigl [}2\varphi -1-\varphi \alpha {\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\varphi +1-\alpha {\bigr ]},&\pm \,2{\bigl [}\varphi -1{\bigr ]},&\pm {\bigl [}1-\varphi \alpha {\bigr ]}&{\Bigr )},\end{array}}}](../I/52353b9fb1cd20ed7de73812fab451bc95ac4c1b.svg)
where is the golden ratio and
See also
References
- ↑ Maeder, Roman. "72: small retrosnub icosicosidodecahedron". MathConsult.
- ↑ Birrell, Robert J. (May 1992). The Yog-sothoth: analysis and construction of the small inverted retrosnub icosicosidodecahedron (M.S.). California State University.
- ↑ Bowers, Jonathan (2000). "Uniform Polychora" (PDF). In Reza Sarhagi (ed.). Bridges 2000. Bridges Conference. pp. 239–246.
External links
- Weisstein, Eric W. "Small retrosnub icosicosidodecahedron". MathWorld.
- Klitzing, Richard. "3D star small retrosnub icosicosidodecahedron".