In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal non-prismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these figures that does not contain triangles or squares. The icosahedron has twenty triangular faces that **truncate** once, and the triangles become hexagons, with pentagons appearing below the pyramids eliminated in the truncation. This is the “soccer ball shape” familiar to millions of people.

The truncated icosahedron is the polyhedron uniform dimensional produced by **truncation** of an icosahedron. Each vertex of the icosahedron becomes a pentagonal face in the truncated icosahedron and each triangular face becomes a hexagonal face. Rectified truncated icosahedron — Craig Kaplan. If a set of the 13 Archimedean solids were constructed with all equal edge lengths, the truncated icosidodecahedron would be the largest.

If you take this figure and **truncate** it again, the twenty hexagons become twenty dodecagons, the twelve pentagons each turn into decagons, and sixty isosceles triangles appear below the pyramids eliminated by this second truncation. The truncated icosidodecahedron is the convex envelope of a rhombicosidodecahedron with cuboids above its 30 squares, whose height/base ratio is φ. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated icosidodecahedron is a zonohedron. Within icosahedral symmetry there are unlimited geometric variations of the truncated icosidodecahedron with isogonal faces.

The truncated icosidodecahedron can also be represented as a spherical mosaic and projected onto the plane by means of a stereographic projection. However, it did not become a familiar form until the introduction in 1970 of the Adidas Telstar soccer ball, whose white hexagons surrounding the black pentagons that form a truncated icosahedron are now iconically associated with the sport of football. The lenses used to focus explosive shock waves from detonators in the Fat Man atomic bomb were constructed in the configuration of a truncated icosahedron (Rhodes 1996, p.). In the mathematical field of graph theory, a truncated icosidodecahedral graph (or large rhombicosidodecahedral graph) is the vertex graph and edges of the truncated icosidodecahedron, one of Archimedes' solids.

After discovering the rectified truncated icosahedron, Kaplan realized that this solid could have been discovered as early as 1568. The truncated icosahedron is also known to chemists as the pure carbon structure known as buckyball (a.