Snub dodecahedron

The snub dodecahedron or snid, also called the snub icosidodecahedron, is one of the 13 Archimedean solids. Its surface consists of 60 snub triangles, 20 more triangles, and 12 pentagons, with four triangles and one pentagon meeting at each vertex. It can be obtained by alternation of the great rhombicosidodecahedron, followed by adjustment of edge lengths to be all equal.

This is one of nine uniform snub polyhedra generated with one set of digonal faces.

Measures
The circumradius R ≈ 2.15584 of the snub dodecahedron with unit edge length is the largest real root of
 * $$4096x^{12}-27648x^{10}+47104x^8-35776x^6+13872x^4-2696x^2+209.$$

Its volume V ≈ 37.61665 is given by the largest real root of
 * $$\begin{align}&2176782336x^{12}-3195335070720x^{10}+162223191936000x^8+1030526618040000x^6\\

{} &+6152923794150000x^4-182124351550575000x^2+187445810737515625.\end{align}$$ These same polynomials define the circumradii and volumes of the great snub icosidodecahedron, the great inverted snub icosidodecahedron, and the great inverted retrosnub icosidodecahedron.

Its dihedral angles may be given as acos(α) for the angle between two triangles, and acos(β) for the angle between a pentagon and a triangle, where α ≈ –0.96210 is the smallest real root of
 * $$729x^6-486x^5-729x^4+756x^3+63x^2-270x+1,$$

and β ≈ –0.89045 is the second to smallest root of
 * $$91125x^{12}-668250x^{10}+2006775x^8-2735100x^6+1768275x^4-502410x^2+43681.$$

Related polyhedra
The disnub icosidodecahedron is a uniform polyhedron compound composed of the two opposite chiral forms of the snub dodecahedron.

It is also related to the icosidodecahedron and small rhombicosidodecahedron through a twisting operation. Twisting the pentagons and triangles of the small rhombicosidodecahedron so the squares become pairs of triangles results in the snub dodecahedron. Continuing the twisting until the triangles become edges results in an icosidodecahedron.