Truncated icosahedron
Truncated icosahedron  

(Click here for rotating model) 

Type  Archimedean solid Uniform polyhedron 
Elements  F = 32, E = 90, V = 60 (χ = 2) 
Faces by sides  12{5}+20{6} 
Schläfli symbols  t{3,5} 
t_{0,1}{3,5}  
Wythoff symbol  2 5  3 
Coxeter diagram  
Symmetry group  I_{h}, H_{3}, [5,3], (*532), order 120 
Rotation group  I, [5,3]^{+}, (532), order 60 
Dihedral Angle  66:138.189685° 65:142.62° 
References  U_{25}, C_{27}, W_{9} 
Properties  Semiregular convex 
Colored faces 
5.6.6 (Vertex figure) 
Pentakis dodecahedron (dual polyhedron) 
Net 
In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose faces are two or more types of regular polygons.
It has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges.
It is the Goldberg polyhedron G_{V}(1,1), containing pentagonal and hexagonal faces.
This geometry is associated with footballs (soccer balls) typically patterned with white hexagons and black pentagons. Geodesic domes are often based on this structure. And it also corresponds to the geometry of the "Bucky Ball" (Carbon60, or C_{60}) molecule.
Contents
Construction
This polyhedron can be constructed from an icosahedron with the 12 vertices truncated (cut off) such that one third of each edge is cut off at each of both ends. This creates 12 new pentagon faces, and leaves the original 20 triangle faces as regular hexagons. Thus the length of the edges is one third of that of the original edges.
Icosahedron 
Cartesian coordinates
Cartesian coordinates for the vertices of a truncated icosahedron centered at the origin are all even permutations of:
 (0, ±1, ±3φ)
 (±2, ±(1+2φ), ±φ)
 (±1, ±(2+φ), ±2φ)
where φ = (1 + √5) / 2 is the golden mean. Using φ^{2} = φ + 1 one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to 9φ + 10. The edges have length 2.^{1}
Orthogonal projections
The truncated icosahedron has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: hexagonal and pentagonal. The last two correspond to the A_{2} and H_{2} Coxeter planes.
Centered by  Vertex  Edge 56 
Edge 66 
Face Hexagon 
Face Pentagon 

Image  
Projective symmetry 
[2]  [2]  [2]  [6]  [10] 
Dimensions
If the edge length of a truncated icosahedron is a, the radius of a circumscribed sphere (one that touches the truncated icosahedron at all vertices) is:
where φ is the golden ratio.
This result is easy to get by using one of the three orthogonal golden rectangles drawn into the original icosahedron (before cut off) as the starting point for our considerations. The angle between the segments joining the center and the vertices connected by shared edge (calculated on the basis of this construction) is approx. 23.281446°.
Area and volume
The area A and the volume V of the truncated icosahedron of edge length a are:
Geometric relations
The truncated icosahedron easily verifies the Euler characteristic:
 32 + 60 − 90 = 2.
With unit edges, the surface area is (rounded) 21 for the pentagons and 52 for the hexagons, together 73 (see areas of regular polygons).
Applications
The balls used in association football and team handball are perhaps the bestknown example of a spherical polyhedron analog to the truncated icosahedron, found in everyday life.^{2} The ball comprises the same pattern of regular pentagons and regular hexagons, but it is more spherical due to the pressure of the air inside and the elasticity of the ball. This ball type was introduced to the World Cup in 1970 (starting in 2006, this iconic design has been superseded by alternative patterns).
Geodesic domes are typically based on this geometry with example structures found across the world, popularized by Buckminster Fuller.
A variation of the icosahedron was used as the basis of the honeycomb wheels (made from a polycast material) used by the Pontiac Motor Division between 1971 to 1976 on its Trans Am and Grand Prix.
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 bombs.^{3}
The truncated icosahedron can also be described as a model of the Buckminsterfullerene (fullerene) (C_{60}), or "buckyball," molecule, an allotrope of elemental carbon, discovered in 1985. The diameter of the football and the fullerene molecule are 22 cm and ca. 1 nm, respectively, hence the size ratio is 220,000,000:1.
Truncated icosahedra in the arts
A truncated icosahedron with "solid edges" is a drawing by Lucas Pacioli illustrating The Divine Proportion.
Related polyhedra
Symmetry: [5,3], (*532)  [5,3]^{+}, (532)  

{5,3}  t{5,3}  r{5,3}  2t{5,3}=t{3,5}  2r{5,3}={3,5}  rr{5,3}  tr{5,3}  sr{5,3} 
Duals to uniform polyhedra  
V5.5.5  V3.10.10  V3.5.3.5  V5.6.6  V3.3.3.3.3  V3.4.5.4  V4.6.10  V3.3.3.3.5 
Symmetry *n42 [n,3] 
Spherical  Euclidean  Compact hyperbolic  Paracompact  

*232 [2,3] D_{3h} 
*332 [3,3] T_{d} 
*432 [4,3] O_{h} 
*532 [5,3] I_{h} 
*632 [6,3] P6m 
*732 [7,3] 
*832 [8,3]... 
*∞32 [∞,3] 

Order  12  24  48  120  ∞  
Truncated figures 
2.6.6 
3.6.6 
4.6.6 
5.6.6 
6.6.6 
7.6.6 
8.6.6 
∞.6.6 
Coxeter Schläfli 
t{3,2} 
t{3,3} 
t{3,4} 
t{3,5} 
t{3,6} 
t{3,7} 
t{3,8} 
t{3,∞} 
Uniform dual figures  
nkis figures 
V2.6.6 
V3.6.6 
V4.6.6 
V5.6.6 
V6.6.6 
V7.6.6 
V8.6.6 
V∞.6.6 
Coxeter 
These uniform starpolyhedra, and one icosahedral stellation have nonuniform truncated icosahedra convex hulls:
Nonuniform truncated icosahedron 2 5  3 
U37 2 5/2  5 
U61 5/2 3  5/3 
U67 5/3 3  2 
U73 2 5/3 (3/2 5/4) 
Complete stellation 

Nonuniform truncated icosahedron 2 5  3 
U38 5/2 5  2 
U44 5/3 5  3 
U56 2 3 (5/4 5/2)  

Nonuniform truncated icosahedron 2 5  3 
U32  5/2 3 3 
See also
 Dodecahedron
 Icosidodecahedron
 Truncated dodecahedron
 truncated rhombic triacontahedron
 hyperbolic soccerball
 fullerene
Notes
 ^ Weisstein, Eric W., "Icosahedral group", MathWorld.
 ^ Kotschick, Dieter (2006). "The Topology and Combinatorics of Soccer Balls". American Scientist 94 (4): 350–357.
 ^ Rhodes, Richard (1996). Dark Sun: The Making of the Hydrogen Bomb. Touchstone Books. p. 195. ISBN 0684824140.
References
 Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 048623729X. (Section 39)
 Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0521554322.
External links
Look up truncated icosahedron in Wiktionary, the free dictionary. 
 Eric W. Weisstein, Truncated icosahedron (Archimedean solid) at MathWorld
 Richard Klitzing, 3D convex uniform polyhedra, x3x5o  ti
 Editable printable net of a truncated icosahedron with interactive 3D view
 The Uniform Polyhedra
 Virtual Reality Polyhedra The Encyclopedia of Polyhedra

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