| Truncated icosahedron | |
|---|---|
 | |
| Type | Archimedean solid Uniform polyhedron Goldberg polyhedron |
| Faces | 32 |
| Edges | 90 |
| Vertices | 60 |
| Symmetry group | Icosahedral symmetry |
| Dual polyhedron | Pentakis dodecahedron |
| Vertex figure | |
 | |
| Net | |
 | |
Ingeometry, thetruncated icosahedron is a polyhedron that can be constructed bytruncating all of theregular icosahedron's vertices. Intuitively, it may be regarded asfootballs (or soccer balls) that are typically patterned with white hexagons and black pentagons. It can be found in the application ofgeodesic dome structures such as those whose architectureBuckminster Fuller pioneered are often based on this structure. It is an example of anArchimedean solid, as well as aGoldberg polyhedron.
The truncated icosahedron can be constructed from aregular icosahedron by cutting off all of its vertices, known astruncation. Each of the 12 vertices at the one-third mark of each edge creates 12 pentagonal faces and transforms the original 20 triangle faces into regular hexagons.[1] Therefore, the resulting polyhedron has 32 faces, 90 edges, and 60 vertices.[2] AGoldberg polyhedron is one whose faces are 12 pentagons and some multiple of 10 hexagons. There are three classes of Goldberg polyhedron, one of them is constructed by truncating all vertices repeatedly, and the truncated icosahedron is one of them, denoted as.[3]
The surface area and the volume of the truncated icosahedron of edge length are:[2]Thesphericity of a polyhedron describes how closely a polyhedron resembles asphere. It can be defined as the ratio of the surface area of a sphere with the same volume to the polyhedron's surface area, from which the value is between 0 and 1. In the case of a truncated icosahedron, it is:[2]
Thedihedral angle of a truncated icosahedron between adjacent hexagonal faces is approximately 138.18°, and that between pentagon-to-hexagon is approximately 142.6°.[4]
The truncated icosahedron is anArchimedean solid, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex.[5] It has the same symmetry as the regular icosahedron, theicosahedral symmetry, and it also has the property ofvertex-transitivity.[6][7] The polygonal faces that meet for every vertex are one pentagon and two hexagons, and thevertex figure of a truncated icosahedron is. The truncated icosahedron's dual ispentakis dodecahedron, aCatalan solid,[8] shares the same symmetry as the truncated icosahedron.[9]
According toSteinitz's theorem, theskeleton of a truncated icosahedron, like that of anyconvex polyhedron, can be represented as apolyhedral graph, meaning aplanar graph (one that can be drawn without crossing edges) and3-vertex-connected graph (remaining connected whenever two of its vertices are removed).[10] The graph is known astruncated icosahedral graph, and it has 60vertices and 90 edges. It is anArchimedean graph because it resembles one of the Archimedean solids. It is acubic graph, meaning that each vertex is incident to exactly three edges.[11][12][13]
The balls used inassociation football andteam handball are perhaps the best-known example of aspherical polyhedron analog to the truncated icosahedron, found in everyday life.[14] The ball comprises the same pattern of regular pentagons and regular hexagons, each of which is painted in black and white respectively; still, its shape is more spherical. It was introduced byAdidas, which debuted theTelstar ball duringWorld Cup in 1970.[15] However, it was superseded in2006.[16]
Geodesic domes are typically based on triangular facetings of this geometry with example structures found across the world, popularized byBuckminster Fuller. An example can be found in the model of abuckminsterfullerene, a truncated icosahedron-shaped geodesic domeallotrope of elemental carbon discovered in 1985.[17] In other engineering and science applications, its shape was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in boththe gadget andFat Manatomic bombs.[18] Its structure can also be found in theprotein ofclathrin.[13]
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The truncated icosahedron was known toArchimedes, who classified the 13 Archimedean solids in a lost work. All that is now known of his work on these shapes comes fromPappus of Alexandria, who merely lists the numbers of faces for each: 12 pentagons and 20 hexagons, in the case of the truncated icosahedron. The first known image and complete description of a truncated icosahedron are from a rediscovery byPiero della Francesca, in his 15th-century bookDe quinque corporibus regularibus, which included five of the Archimedean solids (the five truncations of the regular polyhedra).[19] The same shape was depicted byLeonardo da Vinci, in his illustrations forLuca Pacioli's plagiarism of della Francesca's book in 1509. AlthoughAlbrecht Dürer omitted this shape from the other Archimedean solids listed in his 1525 book on polyhedra,Underweysung der Messung, a description of it was found in his posthumous papers, published in 1538.Johannes Kepler later rediscovered the complete list of the 13 Archimedean solids, including the truncated icosahedron, and included them in his 1609 book,Harmonices Mundi.[20]
