Small retrosnub icosicosidodecahedron

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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 Template:Math. 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

Let $\xi =-\frac{3}{2}-\frac{1}{2}\sqrt{1+4\varphi }\approx -2.86676039917$ be the smallest (most negative) zero of the polynomial $P=x^2+3x+{\varphi }^{-2}$, where $\varphi$ is the golden ratio. Equivalently, $\xi =-1-\sqrt{\varphi +\sqrt{\varphi +\sqrt{\varphi +\cdots }}}=-1-\beta$ where $\beta \approx 1.86676039917$ (Template:OEIS2C) is a root of ${\beta }^2-\beta -\varphi =0.$ Let the point $p$ be given by

$p=\left(\begin{array}{c}{\varphi }^{-1}\xi +{\varphi }^{-3}\\ \xi \\ {\varphi }^{-2}\xi +{\varphi }^{-2}\end{array}\right)$.

Let the matrix $M$ be given by

$M=\left(\begin{array}{ccc}1/2& -\varphi /2& 1/(2\varphi )\\ \varphi /2& 1/(2\varphi )& -1/2\\ 1/(2\varphi )& 1/2& \varphi /2\end{array}\right)$.

$M$ is the rotation around the axis $(1,0,\varphi )$ by an angle of $2\pi /5$, counterclockwise. Let the linear transformations $T_0,\dots ,T_{11}$ 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 $T_i{M}^j$ $(i=0,\dots ,11$, $j=0,\dots ,4)$ 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 $\sqrt{-4\xi -{\varphi }^{-2}}$, and the midradius equals $\sqrt{-\xi }$.

For a small snub icosicosidodecahedron whose edge length is 1, the circumradius is

$R=\frac{1}{2}\sqrt{\frac{\xi -1}{\xi }}\approx 0.5806948001339209$

Its midradius is

$r=\frac{1}{2}\sqrt{\frac{-1}{\xi }}\approx 0.2953073837589815$

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

See also

• List of uniform polyhedra
• Small snub icosicosidodecahedron

External links

• Template:Mathworld
• Template:KlitzingPolytopes

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