Icosidodecahedron

The icosidodecahedron, or id, is a quasiregular polyhedron and one of the 13 Archimedean solids. It consists of 20 equilateral triangles and 12 pentagons, with two of each joining at a vertex. It can be derived as a rectified dodecahedron or icosahedron.

Vertex coordinates
An icosidodecahedron of side length 1 has vertex coordinates given by all permutations of and even permutations of
 * $$\left(±\frac{1+\sqrt5}{2},\,0,\,0\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac12\right).$$

The first set of vertices corresponds to a scaled octahedron which can be inscribed into the icosidodecahedron.

Representations
An icosidodecahedron can be represented by the following Coxeter diagrams:


 * o5x3o (full symmetry)
 * xoxfo5ofxox&#xt (H2 axial, as pentagonal gyrobirotunda, triangle-first)
 * oxFfofx3xfofFxo&#xt (A2 axial)
 * VooFxf oVofFx ooVxfF&#zx (A1×A1×A1 symmetry)
 * oxFf(oV)fFxo ofxF(Vo)Fxfo&#xt (A1×A1 axial)

Related polyhedra
The icosidodecahedron is the colonel of a three-member regiment that also includes the small icosihemidodecahedron and the small dodecahemidodecahedron.

The icosidodecahedron can be split along an equatorial decagonal section to produce two pentagonal rotundas. Since the bases of these rotundas are in opposite orientations, an icosidodecahedron can be called the pentagonal gyrobirotunda. If one rotunda is rotated 36°, so that triangles connect to triangles and pentagons connect to pentagons, the result is the pentagonal orthobirotunda. If a decagonal prism is inserted between the two halves of the icosidodecahedron, the result is the elongated pentagonal gyrobirotunda.