In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U<sub>46</sub>. It has 104 faces (80 triangles, 12 pentagons, and 12 pentagrams), 180 edges, and 60 vertices. As the name indicates, it belongs to the family of snub polyhedra.
Let be the real zero of the polynomial . The number is known as the plastic ratio. Denote by 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 snub icosidodecadodecahedron. The edge length equals , the circumradius equals , and the midradius equals .
For a snub icosidodecadodecahedron whose edge length is 1, the circumradius is
Its midradius is
The medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron.