Truncated icosahedron
Truncated icosahedron  

(Click here for rotating model)  
Type  Archimedean solid Uniform polyhedron 
Elements  F = 32, E = 90, V = 60 ( = 2) 
Faces by sides  12{5}+20{6} 
Conway notation  tI 
Schlfli 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. It also corresponds to the geometry of the "Bucky Ball" (Carbon60, or C_{60}) molecule.
It is used in the celltransitive hyperbolic spacefilling tessellation, the bitruncated order5 dodecahedral honeycomb.
Contents
Construction[edit]
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[edit]
Cartesian coordinates for the vertices of a truncated icosahedron centered at the origin are all even permutations of:
 (0, 1, 3)
 (2, (1+2), )
 (1, (2+), 2)
where = (1 + 5) / 2 is the golden mean. Using ^{2} = + 1 one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to 9 + 10. The edges have length 2.^{[1]}
Permutations:
 X axis
 (3, 0, 1)
 ((1+2), , 2)
 ((2+), 2, 1)
 Y axis
 (1, 3, 0)
 (2, (1+2), )
 (1, (2+), 2)
 Z axis
 (0, 1, 3)
 (, 2, (1+2))
 (2, 1, (2+))
Orthogonal projections[edit]
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] 
Dual 
Spherical tiling[edit]
The truncated icosahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
pentagoncentered  hexagoncentered  
Orthographic projection  Stereographic projections 

Dimensions[edit]
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 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[edit]
The area A and the volume V of the truncated icosahedron of edge length a are:
Geometric relations[edit]
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[edit]
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).^{[citation needed]}
Geodesic domes are typically based on triangular facetings of this geometry with example structures found across the world, popularized by Buckminster Fuller.^{[citation needed]}
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.^{[citation needed]}
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 about 1 nm, respectively, hence the size ratio is 220,000,000:1.^{[citation needed]}
Truncated icosahedra in the arts[edit]
A truncated icosahedron with "solid edges" by Leonardo da Vinci appears as an illustration in Luca Pacioli's book De divina proportione.
Related polyhedra[edit]
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 
Sym. *n42 [n,3]  Spherical  Euclid.  Compact hyperb.  Parac.  Noncompact hyperbolic  

*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]  [12i,3]  [9i,3]  [6i,3]  [3i,3]  
Figures  
Schlfli  t{3,2}  t{3,3}  t{3,4}  t{3,5}  t{3,6}  t{3,7}  t{3,8}  t{3,}  t{3,12i}  t{3,9i}  t{3,6i}  t{3,3i} 
Coxeter  
Uniform dual figures  
nkis figures Config.  V2.6.6  V3.6.6  V4.6.6  V5.6.6  V6.6.6  V7.6.6  V8.6.6  V.6.6  V12i.6.6  V9i.6.6  V6i.6.6  V3i.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 
Truncated icosahedral graph[edit]
Truncated icosahedral graph  

6fold symmetry schlegel diagram  
Vertices  60 
Edges  90 
Automorphisms  120 
Chromatic number  2 
Properties  Cubic, Hamiltonian, regular, zerosymmetric 
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.^{[4]}
5fold symmetry  5fold Schlegel diagram 
See also[edit]
Notes[edit]
 ^ Weisstein, Eric W., "Icosahedral group", MathWorld.
 ^ Kotschick, Dieter (2006). "The Topology and Combinatorics of Soccer Balls". American Scientist 94 (4): 350357. doi:10.1511/2006.60.350.
 ^ Rhodes, Richard (1996). Dark Sun: The Making of the Hydrogen Bomb. Touchstone Books. p. 195. ISBN 0684824140.
 ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
References[edit]
 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. 7986 Archimedean solids. ISBN 0521554322.
External links[edit]
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

