Truncated icosahedron

Structure and basic properties

Important numerical data are simple and widely used: 32 faces (12 pentagons + 20 hexagons), 90 edges and 60 vertices, which satisfy Euler's formula V − E + F = 2 (60 − 90 + 32 = 2). Each vertex has the configuration 5.6.6, meaning one pentagon and two hexagons meet at every vertex. The polyhedron inherits the full icosahedral symmetry of its parent, the regular icosahedron; that symmetry is commonly described as icosahedral symmetry (rotational group of order 60, full group of order 120). The truncated icosahedron can be obtained by truncating the vertices of a regular icosahedron, often denoted by the truncation notation t{3,5} in polyhedral notation.

Construction and geometry

Truncation means slicing off each corner of the original icosahedron at a plane that cuts each incident edge at the same fraction of its length. The original triangular faces become hexagons because each triangle's three corners are removed and replaced by additional edges; the five triangles that met at a vertex produce a pentagonal face in its place. The resulting faces are regular polygons when truncation is performed at the appropriate depth, so the shape is a uniform polyhedron. Geometrically it provides a good approximation to a sphere while remaining composed of flat polygonal faces.

History, names and notation

The name simply records the operation (truncation) applied to the icosahedron. It appears in classical lists of semiregular solids and in later systematic studies of uniform polyhedra. In notation used by geometers the shape is sometimes written as t{3,5} or called the truncated icosahedron; individual face types are often referred to by their polygon names, e.g. pentagons (pentagon) and hexagons (hexagon).

Uses and cultural significance

The truncated icosahedron is best known visually as the pattern used on many traditional soccer balls: a patchwork of pentagons and hexagons. In chemistry the same geometry describes the carbon-60 molecule, buckminsterfullerene (C60), whose atoms occupy the vertices and whose bonds follow the polyhedron's edges. Architects and designers use truncated icosahedral geometry for geodesic-like structures and tilings because it combines aesthetic appeal with structural regularity. It also serves as a standard example in education for discussing symmetry, truncation, and uniform polyhedra.

Notable facts and distinctions

• The truncated icosahedron is uniform but not regular: it has more than one kind of face but all vertices are equivalent.
• Each vertex joins one pentagon and two hexagons (5.6.6), and the figure is one of the 13 classic Archimedean solids.
• It is frequently used when a spherical approximation is needed with a small number of flat panels; its combination of pentagons and hexagons is shared by several natural and engineered structures.