| Truncated Icosahedron |
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| uniform polyhedra | |
| archimedean solids | |
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The truncated icosahedron is an Archimedean Solid . It comprises 12 regular Pentagonal faces, 20 regular Hexagonal faces, 60 vertices and 90 edges. This polyhedron can be constructed from an Icosahedron with the 12 vertices Truncated (cut off), creating 12 new pentagon faces, and leaving the original 20 triangle faces as hexagons. __TOC__ CANONICAL COORDINATES Canonical coordinates for the vertices of a ''truncated icosahedron'' centered at the origin are the Orthogonal rectangles (0,±1,±3φ), (±1,±3φ,0), (±3φ,0,±1) and the orthogonal Cuboid s (±2,±(1+2φ),±φ), (±(1+2φ),±φ,±2), (±φ,±2,±(1+2φ)) along with the orthogonal cuboids (±1,±(2+φ),±2φ), (±(2+φ),±2φ,±1), (±2φ,±1,±(2+φ)), where φ = (1+√5)/2 is the Golden Mean . Using φ2 = φ + 1 one verifies that all vertices are on sphere, centered at the origin, with the radius squared equal to 9φ + 10. GEOMETRIC RELATIONS The truncated icosahedron easily verifies the Euler Characteristic : :32 + 60 - 90 = 2. USES A Football (soccer ball) comprises the same pattern of regular pentagons and regular hexagons, but is more spherical due to the pressure of the air inside. This shape was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in both The Gadget and Fat Man Atomic Bomb s (Richard Rhodes. ''Dark Sun: The Making of the Hydrogen Bomb'', ISBN 0684824140. Touchstone Books, 1996., p. 195). The truncated icosahedron can also be described as a model of the Buckminsterfullerene (C60) molecule. The diameter of the football and the Buckminsterfullerene molecule are 22 cm and ca. 1 Nm , respectively, hence the size ratio is 200,000,000 : 1. ]] TRUNCATED ICOSAHEDRA IN THE ARTS A truncated icosahedron with "solid edges" is a drawing by Lucas Pacioli illustrating ''The Divine Proportion''. SEE ALSO EXTERNAL LINKS
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