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 $U72$. 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.

Cartesian coordinates
Let $$\xi=-\frac32-\frac12\sqrt{1+4\phi}\approx -2.866760399173862$$ be the smallest (most negative) zero of the polynomial $$P=x^2+3x+\phi^{-2}$$, where $$\phi$$ is the golden ratio. 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}$$.

For a small snub icosicosidodecahedron whose edge length is 1, the circumradius is
 * $$R = \frac12\sqrt{\frac{\xi-1}{\xi}} \approx 0.5806948001339209$$

Its midradius is
 * $$r = \frac12\sqrt{\frac{-1}{\xi}} \approx 0.2953073837589815$$

The other zero of $$P$$ plays a similar role in the description of the small snub icosicosidodecahedron.