Great snub dodecicosidodecahedron

Let the point p be given by :p= \begin{pmatrix} \phi^{-\frac12} \\ 0 \\ \phi^{-1} \end{pmatrix} , where \phi is the golden ratio. Let the matrix M be given by :M= \begin{pmatrix} 1/2 & -\phi/2 & 1/(2\phi) \\ \phi/2 & 1/(2\phi) & -1/2 \\ 1/(2\phi) & 1/2 & \phi/2 \end{pmatrix} . M is the rotation around the axis (1, 0, \phi) by an angle of 2\pi/5, counterclockwise. Let the linear transformations T_0, \ldots, T_{11} be the transformations which send a point (x, y, z) to the even permutations of (\pm x, \pm y, \pm z) 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 (i = 0,\ldots, 11, j = 0,\ldots, 4) constitute the group of rotational symmetries of a regular icosahedron. Then the 60 points T_i M^j p are the vertices of a great snub dodecicosidodecahedron. The edge length equals \sqrt2, the circumradius equals 1, and the midradius equals \frac12\sqrt2. For a great snub dodecicosidodecahedron whose edge length is 1, the circumradius is :R = \frac12\sqrt2. Its midradius is :r=\frac12. == Related polyhedra ==

It shares its vertices and edges, as well as 20 of its triangular faces and all its pentagrammic faces, with the great dirhombicosidodecahedron, (although the latter has 60 edges not contained in the great snub dodecicosidodecahedron). It shares its other 60 triangular faces (and its pentagrammic faces again) with the great disnub dirhombidodecahedron. The edges and triangular faces also occur in the compound of twenty octahedra. In addition, 20 of the triangular faces occur in one enantiomer of the compound of twenty tetrahemihexahedra, and the other 60 triangular faces occur in the other enantiomer. == Gallery ==