Truncated icosahedron

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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}
Wythoff symbol 2 5 | 3
Coxeter diagram CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Symmetry group Ih, H3, [5,3], (*532), order 120
Rotation group I, [5,3]+, (532), order 60
Dihedral angle 6-6: 138.189685°
6-5: 142.62°
References U25, C27, W9
Properties Semiregular convex
Truncated icosahedron.png
Colored faces
Truncated icosahedron vertfig.png
(Vertex figure)
Pentakis dodecahedron
(dual polyhedron)
Truncated icosahedron flat-2.svg

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 GPV(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 C60 ("buckyball") molecule.

It is used in the cell-transitive hyperbolic space-filling tessellation, the bitruncated order-5 dodecahedral honeycomb.


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

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)
(±2, ±(1 + 2φ), ±φ)

where φ = 1 + 5/2 is the golden mean. Using φ2 = φ + 1 one verifies that all vertices are on a sphere, centered at the origin, with the radius 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 A2 and H2 Coxeter planes.

Orthogonal projections
Centered by Vertex Edge
Image Dodecahedron t12 v.png Dodecahedron t12 e56.png Dodecahedron t12 e66.png Icosahedron t01 A2.png Icosahedron t01 H3.png
[2] [2] [2] [6] [10]
Dual Dual dodecahedron t01 v.png Dual dodecahedron t01 e56.png Dual dodecahedron t01 e66.png Dual dodecahedron t01 A2.png Dual dodecahedron t01 H3.png

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.

Uniform tiling 532-t12.png Truncated icosahedron stereographic projection pentagon.png
Truncated icosahedron stereographic projection hexagon.png
Orthographic projection Stereographic projections


Mutually orthogonal golden rectangles drawn into the original icosahedron (before cut off)

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.


The balls used in association football and team handball are perhaps the best-known 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) (C60), 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

A truncated icosahedron with "solid edges" by Leonardo da Vinci appears as an illustration in Luca Pacioli's book De divina proportione.


Related polyhedra

These uniform star-polyhedra, and one icosahedral stellation have nonuniform truncated icosahedra convex hulls:

Truncated icosahedral graph

Truncated icosahedral graph
Truncated icosahedral graph.png
6-fold symmetry schlegel diagram
Vertices 60
Edges 90
Automorphisms 120
Chromatic number 3
Properties Cubic, Hamiltonian, regular, zero-symmetric

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]

Orthographic projection
Icosahedron t01 H3.png
5-fold symmetry
Truncated icosahedral graph pentcenter.png
5-fold Schlegel diagram


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


  1. ^ Weisstein, Eric W. "Icosahedral group". MathWorld. 
  2. ^ Kotschick, Dieter (2006). "The Topology and Combinatorics of Soccer Balls". American Scientist. 94 (4): 350–357. doi:10.1511/2006.60.350. 
  3. ^ Rhodes, Richard (1996). Dark Sun: The Making of the Hydrogen Bomb. Touchstone Books. p. 195. ISBN 0-684-82414-0. 
  4. ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 268 
  5. ^ Weisstein, Eric W. "Truncated icosahedral graph". MathWorld. 
  6. ^ Godsil, C. and Royle, G. Algebraic Graph Theory New York: Springer-Verlag, p. 211, 2001
  7. ^ Kostant, B. The Graph of the Truncated Icosahedron and the Last Letter of Galois. Notices Amer. Math. Soc. 42, 1995, pp. 959-968 PDF
  8. ^ 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. 
  9. ^ Durer, A. (1471–1528). "German artist who made an early model of a regular truncated icosahedron". 
  10. ^ Dresselhaus, M. S.; Dresselhaus, G.; Eklund, P. C. (1996). Science of fullerenes and carbon nanotubes. San Diego, CA: Academic Press. ISBN 012-221820-5. 


  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  (Section 3-9)
  • Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2. 

External links

  • Eric Wolfgang 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
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