Snub dodecadodecahedron
In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as. It has 84 faces, 150 edges, and 60 vertices. It is given a Schläfli symbol as a snub great dodecahedron.
Cartesian coordinates
Let be the smallest real zero of the polynomial. Denote by the golden ratio. Let the point be given byLet 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