Small retrosnub icosicosidodecahedron explained

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 . It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices.^[1] It is given a Schläfli symbol sr.

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.

Cartesian coordinates

Let

\xi =- (32- (12 \sqrt{ 1+4 \phi} ≈)/(-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 byp = \begin{pmatrix} \phi ^{-1} \xi + \phi ^{-3} \\ \xi \\ \phi ^{-2} \xi + \phi ^{-2} \end{pmatrix} .Let the matrix M be given byM = \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 = (12 \sqrt{ (\xi -1)/(\xi))/() } \approx 0.5806948001339209Its midradius isr = (12 \sqrt{ (-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

Notes and References

1. Web site: 72: small retrosnub icosicosidodecahedron. Maeder. Roman. MathConsult.
2. M.S. . Birrell . Robert J. . May 1992 . The Yog-sothoth: analysis and construction of the small inverted retrosnub icosicosidodecahedron . California State University.
3. Uniform Polychora . Bowers . Jonathan . 2000 . Reza Sarhagi . Bridges 2000 . 239–246 . Bridges Conference .