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} 
Conway notation  tI 
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 GP_{V}(1,1) or {5+,3}_{1,1}, containing pentagonal and hexagonal faces.
This geometry is associated with footballs typically patterned with white hexagons and black pentagons. Geodesic domes such as those whose architecture Buckminster Fuller pioneered are often based on this structure. It also corresponds to the geometry of the fullerene C_{60} ("buckyball") 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.
Cartesian coordinates[edit]
Cartesian coordinates for the vertices of a truncated icosahedron centered at the origin are all even permutations of:
 (0, ±1, ±3φ)
 (±1, ±(2 + φ), ±2φ)
 (±φ, ±2, ±φ^{3})
where φ = 1 + √5/2 is the golden mean. The circumradius is √9φ + 10 and the edges have length 2.^{[1]}
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 

Solid  
Wireframe  
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 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 approximately 23.281446°.
Area and volume[edit]
The area A and the volume V of the truncated icosahedron of edge length a are:
With unit edges, the surface area is (rounded) 21 for the pentagons and 52 for the hexagons, together 73 (see areas of regular polygons).
The truncated icosahedron easily demonstrates the Euler characteristic:
 32 + 60 − 90 = 2.
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).
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 and 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 0.71 nm, respectively, hence the size ratio is ≈31,000,000:1.
In popular craft culture, large sparkleballs can be made using a icosahedron pattern and plastic, styrofoam or paper cups.
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.
Gallery[edit]
The truncated icosahedron (left) compared to an association football.
Fullerene C_{60} molecule
Truncated icosahedral radome on a weather station
A wooden truncated icosahedron artwork by George W. Hart.
Related polyhedra[edit]
Family of uniform icosahedral polyhedra  

Symmetry: [5,3], (*532)  [5,3]^{+}, (532)  
{5,3}  t{5,3}  r{5,3}  t{3,5}  {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 
*n32 symmetry mutation of truncated tilings: n.6.6  

Sym. *n42 [n,3] 
Spherical  Euclid.  Compact  Parac.  Noncompact hyperbolic  
*232 [2,3] 
*332 [3,3] 
*432 [4,3] 
*532 [5,3] 
*632 [6,3] 
*732 [7,3] 
*832 [8,3]... 
*∞32 [∞,3] 
[12i,3]  [9i,3]  [6i,3]  
Truncated figures 

Config.  2.6.6  3.6.6  4.6.6  5.6.6  6.6.6  7.6.6  8.6.6  ∞.6.6  12i.6.6  9i.6.6  6i.6.6  
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 
These uniform starpolyhedra, and one icosahedral stellation have nonuniform truncated icosahedra convex hulls:
Uniform star polyhedra with truncated icosahedra convex hulls  


Truncated icosahedral graph[edit]
Truncated icosahedral graph  

6fold symmetry schlegel diagram  
Vertices  60 
Edges  90 
Automorphisms  120 
Chromatic number  3 
Properties  Cubic, Hamiltonian, regular, zerosymmetric 
Table of graphs and parameters 
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]}^{[5]}^{[6]}^{[7]}
5fold symmetry 
5fold Schlegel diagram 
History[edit]
The truncated icosahedron was known to Archimedes who studied vertextransitive polyhedra. However, that work was lost. Later, Johannes Kepler rediscovered and wrote about these solids, including the truncated icosahedron.
The structure associated was described by Leonardo da Vinci.^{[8]} Albrecht Dürer also reproduced a similar icosahedron containing 12 pentagonal and 20 hexagonal faces but there are no clear documentations of this.^{[9]}^{[10]}
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): 350–357. 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. 268
 ^ Weisstein, Eric W. "Truncated icosahedral graph". MathWorld.
 ^ Godsil, C. and Royle, G. Algebraic Graph Theory New York: SpringerVerlag, p. 211, 2001
 ^ Kostant, B. The Graph of the Truncated Icosahedron and the Last Letter of Galois. Notices Amer. Math. Soc. 42, 1995, pp. 959968 PDF
 ^ Saffaro, L. (1992). "Cosmoids, Fullerenes and continuous polygons". In Taliani, C.; Ruani, G.; Zamboni, R. Proceedings of the First Italian Workshop on Fullerenes: States and Perspectives. 2. Singapore: World Scientific. p. 55. ISBN 9810210825.
 ^ Durer, A. (1471–1528). "German artist who made an early model of a regular truncated icosahedron".
 ^ Dresselhaus, M. S.; Dresselhaus, G.; Eklund, P. C. (1996). Science of fullerenes and carbon nanotubes. San Diego, CA: Academic Press. ISBN 0122218205.
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. 79–86 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.
 Klitzing, Richard. "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
 3D paper data visualization World Cup ball