In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as . It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläfli symbol as a snub great dodecahedron.
Let be the smallest real zero of the polynomial . 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 dodecadodecahedron. The edge length equals , the circumradius equals , and the midradius equals .
For a great snub icosidodecahedron whose edge length is 1, the circumradius is
Its midradius is
The other real root of P plays a similar role in the description of the Inverted snub dodecadodecahedron
The medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.