Small retrosnub icosicosidodecahedron

Short description: Uniform star polyhedron with 112 faces

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}
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

File:Small retrosnub icosicosidodecahedron.stl

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. 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).

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 $$\begin{array}{crrrc}(& \pm [1-\phi -\alpha ],& 0,& \pm [3-\phi \alpha ]& ),\\ (& \pm [\phi -1-\alpha ],& \pm 2,& \pm [2\phi -1-\phi \alpha ]& ),\\ (& \pm [\phi +1-\alpha ],& \pm 2[\phi -1],& \pm [1-\phi \alpha ]& ),\end{array}$$ where $\phi =\frac{1+\sqrt{5}}{2}$ is the golden ratio and $\alpha =\sqrt{3\phi -2}.$

See also

• List of uniform polyhedra
• Small snub icosicosidodecahedron

External links

• Weisstein, Eric W.. "Small retrosnub icosicosidodecahedron". http://mathworld.wolfram.com/SmallRetrosnubIcosicosidodecahedron.html. 
• Klitzing, Richard. "3D star small retrosnub icosicosidodecahedron". https://bendwavy.org/klitzing/dimensions/../incmats/sirsid.htm.