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.

It is one of two chiral Archimedean solids, the other being the snub cube.

Like the snub cube, the triangles come in two types: 20 that only connect to other triangles with rotational triangular symmetry, and 60 snub triangles that connect to the pentagons with no local symmetry.

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
 * $$91125x^{12}-668250x^{10}+2006775x^8-2735100x^6+1768275x^4-502410x^2+43681.$$

Vertex coordinates
The coordinates of a snub dodecahedron, centered at the origin and with unit edge length, are given by all even permutations with an odd number of sign changes of: as well as all even permutations with an even number of sign changes of: where
 * $$\left(\frac{\phi\sqrt{\phi(\xi-1-\frac1\xi)}}2,\,\frac{\xi\phi\sqrt{3-\xi^2}}2,\,\frac{\phi\sqrt{\xi(\xi+\phi)+1}}2\right),$$
 * $$\left(\frac{\phi\sqrt{3-\xi^2}}2,\,\frac{\xi\phi\sqrt{1-\xi+\frac{1+\phi}\xi}}2,\,\frac{\phi\sqrt{\xi(\xi+1)}}2\right),$$
 * $$\left(\frac{\xi^2\phi\sqrt{\phi(\xi-1-\frac1\xi)}}2,\,\frac{\phi\sqrt{\xi+1-\phi}}2,\,\frac{\sqrt{\xi^2(1+2\phi)-\phi}}2\right),$$
 * $$\left(\frac{\xi^2\phi\sqrt{3-\xi^2}}2,\,\frac{\xi\phi\sqrt{\phi(\xi-1-\frac1\xi)}}2,\,\frac{\phi^2\sqrt{\xi(\xi+\phi)+1}}{2\xi}\right),$$
 * $$\left(\frac{\sqrt{\phi(\xi+2)+2}}2,\,\frac{\phi\sqrt{1-\xi+\frac{1+\phi}\xi}}2,\,\frac{\xi\sqrt{\xi(1+\phi)-\phi}}2\right),$$
 * $$\phi = \frac{1+\sqrt5}2,$$
 * $$\xi = \sqrt[3]{\frac{\phi+\sqrt{\phi-\frac5{27}}}2}+\sqrt[3]{\frac{\phi-\sqrt{\phi-\frac5{27}}}2}.$$

Variations
The snub dodecahedron has a general variant that maintains the chiral dodecahedral symmetry. It generally has 12 pentagons of one edge size, 20 triangles of a second size, and 60 scalene triangles using the first two edge sizes along with a third edge size, that joins two scalene triangle faces.

The most notable of these variants is the one obtained as the direct alternation of the uniform great rhombicosidodecahedron. This variant uses 12 pentagons of size $$\sqrt{\frac{5+\sqrt5}{2}}$$, 20 triangles of size $$\sqrt3$$, and 60 scalene triangles, with one of each of these edges along with edges joining them of length $$\sqrt2$$, as faces.

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.