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|Symmetry group||icosahedral (Ih)|
|Dual polyhedron||pentakis dodecahedron|
|Properties||convex, semi-regular (vertex-uniform)|
Canonical coordinates for the vertices of a truncated icosahedron centered at the origin are the orthogonal rectangles (0,±1,±3τ), (±1,±3τ,0), (±3τ,0,±1) and the orthogonal bricks/3D-rectangles (±2,±(1+2τ),±τ), (±(1+2τ),±τ,±2), (±τ,±2,±(1+2τ)) along with the orthogonal bricks/3D-rectangles (±1,±(2+τ),±2τ), (±(2+τ),±2τ,±1), (±2τ,±1,±(2+τ)), where τ = (1+√5)/2 is the golden mean.
It has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges. One easily verifies the Euler characteristic:
- 32 + 60 - 90 = 2.
A football (soccer ball) is like this polyhedron except that it is more spherical, because the faces bulge due 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 the Fat Man atomic bomb (Richard Rhodes. Dark Sun: The Making of the Hydrogen Bomb, ISBN 0684824140. Touchstone Books, 1996., p. 195).
Truncated icosahedra in the arts
A truncated icosahedron with "solid edges" is a drawing by Lucas Pacioli illustrating The Divine Proportion.
- The Uniform Polyhedra
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra
- Paper Models of Polyhedra Many links
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