one icosahedral stellation have nonuniform truncated icosahedra convex hulls: This polyhedron looks similar to the uniform chamfered dodecahedron which has 12 pentagons, but 30 hexagons. Truncated icosahedral graph In the mathematical field of graph theory, a truncated icosahedral graph is the graph of vertices and edges of the truncated icosahedron, one of the Archimedean solids. It has 60 vertices and 90 edges, and is a cubic Archimedean graph. History The truncated icosahedron was known to Archimedes, who classified the 13 Archimedean solids in a lost work. All we know of his work on these shapes comes from Pappus 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 is from a rediscovery by Piero della Francesca, in his 15th-century book De quinque corporibus regularibus, which included five of the Archimedean solids (the five truncations of the regular polyhedra). The same shape was depicted by Leonardo da Vinci, in his illustrations for Luca Pacioli's plagiarism of della Francesca's book in 1509. Although Albrecht 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. See also Fullerene Buckminsterfullerene (C60) Hyperbolic soccerball Snyder equal-area projection Soccer ball Adidas Telstar Notes References (Section 3-9) External links Editable printable net of a truncated icosahedron with interactive 3D view The Uniform Polyhedra "Virtual Reality Polyhedra"—The Encyclopedia of Polyhedra 3D paper data visualization World Cup ball Archimedean solids Goldberg polyhedra Individual graphs Truncated tilings Uniform polyhedra