Small retrosnub icosicosidodecahedron

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}.

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

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

Let $\xi =-{\frac {3}{2}}-{\frac {1}{2}}{\sqrt {1+4\phi }}\approx -2.86676039917$ be the smallest (most negative) zero of the polynomial $P=x^{2}+3x+\phi ^{-2}$ , where $\phi$ is the golden ratio. Equivalently, $\xi =-1-{\sqrt {\phi +{\sqrt {\phi +{\sqrt {\phi +\cdots }}}}}}\,=-1-\beta$ where $\beta \approx 1.86676039917$ (OEIS: A275828) is a root of $\beta ^{2}-\beta -\phi =0.$ Let the point $p$ be given by

p={\begin{pmatrix}\phi ^{-1}\xi +\phi ^{-3}\\\xi \\\phi ^{-2}\xi +\phi ^{-2}\end{pmatrix}}

.

Let the matrix $M$ be given by

M={\begin{pmatrix}1/2&-\phi /2&1/(2\phi )\\\phi /2&1/(2\phi )&-1/2\\1/(2\phi )&1/2&\phi /2\end{pmatrix}}

.

$M$ is the rotation around the axis $(1,0,\phi )$ by an angle of $2\pi /5$ , counterclockwise. Let the linear transformations $T_{0},\ldots ,T_{11}$ be the transformations which send a point $(x,y,z)$ to the even permutations of $(\pm x,\pm y,\pm z)$ with an even number of minus signs. The transformations $T_{i}$ constitute the group of rotational symmetries of a regular tetrahedron. The transformations $T_{i}M^{j}$ $(i=0,\ldots ,11$ , $j=0,\ldots ,4)$ constitute the group of rotational symmetries of a regular icosahedron. Then the 60 points $T_{i}M^{j}p$ are the vertices of a small snub icosicosidodecahedron. The edge length equals $-2\xi$ , the circumradius equals ${\sqrt {-4\xi -\phi ^{-2}}}$ , and the midradius equals ${\sqrt {-\xi }}$ .