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 symbol	t{3,5}
Wythoff symbol	2 5 | 3
Coxeter diagram
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
Colored faces	5.6.6 (Vertex figure)
Pentakis dodecahedron (dual polyhedron)	Net

In geometry, the truncated icosahedron is an Archimedean solid, one of thirteen 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.

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 φ² = 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.

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 5-6	Edge 6-6	Face Hexagon	Face Pentagon
Image
Projective symmetry	[2]	[2]	[2]	[6]	[10]

Dimensions

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:

$r_u = \frac{a}{2} \sqrt{1 + 9\varphi^2} = \frac{a}{4} \sqrt{58 +18\sqrt{5}} \approx 2.47801866 \cdot a$

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:

$\begin{align} A & = \left ( 20 \cdot \frac32\sqrt{3} + 12 \cdot \frac54\sqrt{ 1 + 2/\sqrt{5}} \right ) a^2 \approx 72.607253a^2 \\ V & = \frac{1}{4} (125+43\sqrt{5}) a^3 \approx 55.2877308a^3. \\ \end{align}$

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 truncated icosahedron (left) compared to a soccer ball.

Fullerene C₆₀ molecule

Truncated icosahedral radome on a weather station

The balls used in soccer and team handball are perhaps the best-known example of a spherical polyhedron analog to the truncated icosahedron, found in everyday life. 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 in 1970; starting with the 2006 World Cup, the design has been superseded by newer patterns.

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.

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 soccer ball and the fullerene molecule are 22 cm and ca. 1 nm, respectively, hence the size ratio is 220,000,000:1.

The truncated icosahedron is also hypothesized in geology to be the driving force behind many tectonic fabrics on earth. According to the theory, since the shape is a close geometric analog to the shape of the earth, it can explain the trend of many different fracture and associated features in plate tectonic rifting and craton shape.

Truncated icosahedra in the arts

A truncated icosahedron with "solid edges" is a drawing by Lucas Pacioli illustrating The Divine Proportion.

Related polyhedra

Family of uniform icosahedral polyhedra

Symmetry: [5,3], (*532)							[5,3]+, (532)
{5,3}	t0,1{5,3}	t1{5,3}	t0,1{3,5}	{3,5}	t0,2{5,3}	t0,1,2{5,3}	s{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

Dimensional family of truncated polyhedra and tilings: n.6.6

Symmetry *n42 [n,3]	Spherical				Euclidean	Hyperbolic...
	*232 [2,3] D3h	*332 [3,3] Td	*432 [4,3] Oh	*532 [5,3] Ih	*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	t0,1{3,2}	t0,1{3,3}	t0,1{3,4}	t0,1{3,5}	t0,1{3,6}	t0,1{3,7}	t0,1{3,8}	t0,1{3,∞}
Uniform dual figures
n-kis 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 star-polyhedra, 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

References

• 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)

External links

• 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