Truncated icosahedron in the context of Goldberg polyhedron

A football or soccer ball is the ball used in the sport of association football. The ball's spherical shape, as well as its size, mass, and material composition, are specified by Law 2 of the Laws of the Game maintained by the International Football Association Board. Additional, more stringent standards are specified by FIFA and other big governing bodies for the balls used in the competitions they sanction.

Early footballs began as animal bladders or stomachs that would easily fall apart if kicked too much. Improvements became possible in the 19th century with the introduction of rubber and discoveries of vulcanization by Charles Goodyear. The modern 32-panel ball design was developed in 1962 by Eigil Nielsen, and technological research continues to develop footballs with improved performance. The 32-panel ball design was soon joined by 24-panel balls as well as 42-panel balls, both of which improved on performance prior to 2007.

In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron.

Every polyhedron with icosahedral symmetry has 60 rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120. The full symmetry group is the Coxeter group of type H₃. It may be represented by Coxeter notation [5,3] and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group A₅ on 5 letters.

In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. A polyhedron whose vertices are equidistant from its center can be conveniently studied by projecting its edges onto the sphere to obtain a corresponding spherical polyhedron.

The most familiar spherical polyhedron is the soccer ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the beach ball, thought of as a hosohedron.

Buckminsterfullerene is a type of fullerene with the formula C
₆₀. It has a cage-like fused-ring structure (truncated icosahedron) made of twenty hexagons and twelve pentagons, and resembles a football. Each of its 60 carbon atoms is bonded to its three neighbors.

Buckminsterfullerene is a black solid that dissolves in hydrocarbon solvents to produce a purple solution. The substance was discovered in 1985 and has received intense study, although few real world applications have been found.