Small_retrosnub_icosicosidodecahedron Knowpia

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	I_h, [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 $U 72$ . 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 edit

Its convex hull is a nonuniform truncated dodecahedron.

Truncated dodecahedron	Convex hull	Small retrosnub icosicosidodecahedron

Cartesian coordinates edit

Cartesian coordinates for the vertices of a small retrosnub icosicosidodecahedron are all the even permutations of

{\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}}

where

\varphi ={\tfrac {1+{\sqrt {5}}}{2}}

is the golden ratio and

\alpha ={\sqrt {3\varphi -2}}.

References edit

^ 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.