Truncated icosahedron
In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares.
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  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 
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 (soccer balls) 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.
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. In addition the shape has 1440 interior diagonals
Characteristics
In Geometry and Graph theory, there are some standard polyhedron characteristics.
Cartesian coordinates
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, ±(2φ + 1))
where φ = 1 + √5/2 is the golden mean. The circumradius is √9φ + 10 ≈ 4.956 and 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 

Solid  
Wireframe  
Projective symmetry 
[2]  [2]  [2]  [6]  [10] 
Dual 
Spherical tiling
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
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
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
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.
In the arts
 The truncated icosahedron (left) compared with an association football.
 Fullerene C_{60} molecule
 Truncated icosahedral radome on a weather station
 Truncated icosahedron machined out of 6061T6 aluminum
 A wooden truncated icosahedron artwork by George W. Hart.
Related polyhedra
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
Truncated icosahedral graph  

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
The truncated icosahedron was known to Archimedes, who classified the 13 Archimedean solids in a lost work. All we know of his work on these shapes comes from Pappus of Alexandria, who merely lists the numbers of faces for each: 12 pentagons and 20 hexagons, in the case of the truncated icosahedron. The first known image and complete description of a truncated icosahedron is from a rediscovery by Piero della Francesca, in his 15thcentury book De quinque corporibus regularibus,[8] which included five of the Archimedean solids (the five truncations of the regular polyhedra). The same shape was depicted by Leonardo da Vinci, in his illustrations for Luca Pacioli's plagiarism of della Francesca's book in 1509. Although Albrecht Dürer omitted this shape from the other Archimedean solids listed in his 1525 book on polyhedra, Underweysung der Messung, a description of it was found in his posthumous papers, published in 1538. Johannes Kepler later rediscovered the complete list of the 13 Archimedean solids, including the truncated icosahedron, and included them in his 1609 book, Harmonices Mundi.[9]
See also
Notes
 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. pp. 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
 Katz, Eugene A. (2011). "Bridges between mathematics, natural sciences, architecture and art: case of fullerenes". Art, Science, and Technology: Interaction Between Three Cultures, Proceedings of the First International Conference. pp. 60–71.
 Field, J. V. (1997). "Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler". Archive for History of Exact Sciences. 50 (3–4): 241–289. doi:10.1007/BF00374595 (inactive 31 May 2021). JSTOR 41134110. MR 1457069.CS1 maint: DOI inactive as of May 2021 (link)
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). "Archimedean solids". Polyhedra: "One of the Most Charming Chapters of Geometry". Cambridge: Cambridge University Press. pp. 79–86. ISBN 0521554322. OCLC 180091468.