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
3D model of a medial inverted pentagonal hexecontahedron
The medial inverted pentagonal hexecontahedron (or midly petaloid ditriacontahedron) is a nonconvex isohedralpolyhedron. 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 (least negative) 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 lengthThe dihedral angle equals . The other real zero of the polynomial plays a similar role for the medial pentagonal hexecontahedron.