Small snub icosicosidodecahedron

In geometry, the small snub icosicosidodecahedron or snub disicosidodecahedron is a uniform star polyhedron, indexed as U₃₂. It has 112 faces, 180 edges, and 60 vertices. Its stellation core is a truncated pentakis dodecahedron. It also called a holosnub icosahedron,
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.

Convex hull

Truncated icosahedron

Convex hull

Let be largest zero of the polynomial, where is the golden ratio. 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 retrosnub icosicosidodecahedron.