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. It is given a Schläfli symbol sr{âµ/âÂÂ,ó/âÂÂ}.
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).
Its convex hull is a nonuniform truncated dodecahedron.
Let be the smallest (most negative) zero of the polynomial , where is the golden ratio. Equivalently, where () is a root of Let the point be given by
Let the matrix be given by
is the rotation around the axis by an angle of , counterclockwise. Let the linear transformations be the transformations which send a point to the even permutations of with an even number of minus signs. The transformations constitute the group of rotational symmetries of a regular tetrahedron. The transformations , constitute the group of rotational symmetries of a regular icosahedron. Then the 60 points are the vertices of a small snub icosicosidodecahedron. The edge length equals , the circumradius equals , and the midradius equals .
For a small snub icosicosidodecahedron whose edge length is 1, the circumradius is
Its midradius is
The other zero of plays a similar role in the description of the small snub icosicosidodecahedron.