Inverted snub dodecadodecahedron


In geometry, the inverted snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U60. It is given a Schläfli symbol

Cartesian coordinates

Let be the largest 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 snub dodecadodecahedron.

Related polyhedra

Medial inverted pentagonal hexecontahedron

The medial inverted pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform inverted snub dodecadodecahedron. Its faces are irregular nonconvex pentagons, with one very acute angle.

Proportions

Denote the golden ratio by, and let be the largest real zero of the polynomial. Then each face has three equal angles of, one of and one of. Each face has one medium length edge, two short and two long ones. If the medium length is, then the short edges have length
and the long edges have length
The dihedral angle equals. The other real zero of the polynomial plays a similar role for the medial pentagonal hexecontahedron.