Disnub icosidodecahedron

The disnub icosidodecahedron, dissid, or compound of two snub dodecahedra is a uniform polyhedron compound. It consists of 120 snub triangles, 40 further triangles, and 24 pentagons (the latter two can combine in pairs due to faces in the same plane). Four triangles and one pentagon join at each vertex.

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 ≈ 75.23330 is given by twice 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
 * $$91125x^{12}-668250x^{10}+2006775x^8-2735100x^6+1768275x^4-502410x^2+43681.$$