rhombicosahedra

A rhombicosahedron is a convex polyhedron that is part of the Archimedean and Johnson families of solids. The most common example, the small rhombicosidodecahedron, has 62 faces composed of 20 equilateral triangles, 30 squares, and 12 regular pentagons. It has 120 edges and 60 vertices, with each vertex where two triangles, two squares and one pentagon meet. The solid is vertex‑transitive, meaning all of its vertices are equivalent under its symmetry operations, giving it the symmetry group of the icosahedron (icosahedral symmetry, order 120).

The geometry of the rhombicosidodecahedron can be described using coordinates derived from the golden ratio φ = (1+√5)/2. One convenient set of coordinates for a centered version of the solid has the vertices at all even permutations of (0, ±1, ±φ) and (±1, ±φ, ±φ⁻¹). These coordinates produce a symmetrical shape that can be scaled to an edge length of one or any desired size.

Rhombicosahedra are related to several other well‑known polyhedra. Truncating an icosidodecahedron or an icosahedron in particular ways produces the small rhombicosidodecahedron. Similarly, truncating the icosidodecahedron further yields the great rhombicosidodecahedron, which shares the same vertex arrangement but has larger faces and 80 total faces. The rhombicosidodecahedron also appears as a rectified truncated icosahedron, showing that it can be obtained through multiple truncation and rectification sequences.

These solids are frequently used in architectural design, mathematics education, and as building blocks in molecular modeling. Their high degree of symmetry and mixed face types make them useful for illustrating concepts such as vertex transitivity, Euler's formula, and the relationship between duals and original polyhedra. They are also included in many sets of Platonic and Archimedean solids for enumeration and comparison purposes.