In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U46. 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.

Cartesian coordinates

Let be the real zero of the polynomial x^3-x-1. The number \rho is known as the plastic ratio. Denote by \phi 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 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

Related polyhedra

Medial hexagonal hexecontahedron

The medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron.