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 be the smallest (most negative) zero of the polynomial, where \phi is the golden ratio. Let the point p be given by Let the matrix M be given by M is the rotation around the axis by an angle of 2\pi/5, counterclockwise. Let the linear transformations be the transformations which send a point (x, y, z) to the even permutations of 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, 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, and the midradius equals \sqrt{-\xi}. For a small snub icosicosidodecahedron whose edge length is 1, the circumradius is Its midradius is The other zero of P plays a similar role in the description of the small snub icosicosidodecahedron.