One way to construct the icosidodecahedron is to start with two pentagonal rotunda by attaching them to their bases.

These rotundas cover their decagonal base so that the resulting polyhedron has 32 faces, 30 vertices, and 60 edges.

This construction is similar to one of the Johnson solids, the pentagonal orthobirotunda.

The difference is that the icosidodecahedron is constructed by twisting its rotundas by 36°, a process known as gyration, resulting in the pentagonal face connecting to the triangular one.

[1][2] Convenient Cartesian coordinates for the vertices of an icosidodecahedron with unit edges are given by the even permutations of:

The icosidodecahedron is an Archimedean solid, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex.

Its dual polyhedron is rhombic triacontahedron, a Catalan solid.

[4] Only a few uniform polytopes have this property, including the four-dimensional 600-cell, the three-dimensional icosidodecahedron, and the two-dimensional decagon.

The icosidodecahedron contains 12 pentagons of the dodecahedron and 20 triangles of the icosahedron: The icosidodecahedron exists in a sequence of symmetries of quasiregular polyhedra and tilings with vertex configurations (3.n)2, progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane.

[7][8] The truncated cube can be turned into an icosidodecahedron by dividing the octagons into two pentagons and two triangles.

Eight uniform star polyhedra share the same vertex arrangement.

The vertex arrangement is also shared with the compounds of five octahedra and of five tetrahemihexahedra.

In four-dimensional geometry, the icosidodecahedron appears in the regular 600-cell as the equatorial slice that belongs to the vertex-first passage of the 600-cell through 3D space.

The wireframe figure of the 600-cell consists of 72 flat regular decagons.

Six of these are the equatorial decagons to a pair of opposite vertices, and these six form the wireframe figure of an icosidodecahedron.

If a 600-cell is stereographically projected to 3-space about any vertex and all points are normalised, the geodesics upon which edges fall comprise the icosidodecahedron's barycentric subdivision.

[9] The icosidodecahedron may appear in structures, as in the geodesic dome or the Hoberman sphere.

In Star Trek universe, the Vulcan game of logic Kal-Toh has the goal of creating a shape with two nested holographic icosidodecahedra joined at the midpoints of their segments.