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Truncated icosahedron
Truncated icosahedron
(Click here for rotating model)
Type Archimedean solid
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-Dynkin CDW dot.pngCDW 5.pngCDW ring.pngCDW 3.pngCDW ring.png
Symmetry Ih
or (*532)
References U25, C27, W9
Properties Semiregular convex
Truncated icosahedron color
Colored faces
Truncated icosahedron
5.6.6
(Vertex figure)
Pentakisdodecahedron.jpg
Pentakis dodecahedron
(dual polyhedron)
Truncated icosahedron Net
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 polygon.

It has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges.

Contents

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.png
Icosahedron

Cartesian coordinates

Cartesian coordinates for the vertices of a truncated icosahedron centered at the origin are the orthogonal rectangles (0,±1,±3φ), (±1,±3φ,0), (±3φ,0,±1) and the orthogonal cuboids (±2,±(1+2φ),±φ), (±(1+2φ),±φ,±2), (±φ,±2,±(1+2φ)) along with the orthogonal cuboids (±1,±(2+φ),±2φ), (±(2+φ),±2φ,±1), (±2φ,±1,±(2+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 squared equal to 9φ + 10. The edges have length 2.

Area and volume

The area A and the volume V of the truncated icosahedron of edge length a are:

\begin{align} A & = 3 \left ( 10\sqrt{3} + \sqrt{5} \sqrt{5 + 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.
The fullerene C60 molecule.

The association football (soccer) ball is perhaps the best-known example of a spherical polyhedron analog to the truncated icosahedron, found in everyday life. [1] 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 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. [2]

The truncated icosahedron can also be described as a model of the Buckminsterfullerene (fullerene) (C60), or "buckyball," molecule, a recently-identified allotrope of elemental carbon. 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 the closest 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.[3][4]

Truncated icosahedra in the arts

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

Related polyhedra

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

Nonuniform truncated icosahedron.png
Nonuniform
truncated icosahedron
2 5 |3
Great truncated dodecahedron.png
U37
2 5/2 | 5
Great dodecicosidodecahedron.png
U61
5/2 3 | 5/3
Uniform great rhombicosidodecahedron.png
U67
5/3 3 | 2
Great rhombidodecahedron.png
U73
2 5/3 (3/2 5/4)
Complete icosahedron ortho stella.png
Complete icosahedron
Rhombidodecadodecahedron convex hull.png
Nonuniform
truncated icosahedron
2 5 |3
Rhombidodecadodecahedron.png
U38
5/2 5 | 2
Icosidodecadodecahedron.png
U44
5/3 5 | 3
Rhombicosahedron.png
U56
2 3 (5/4 5/2) |
Small snub icosicosidodecahedron convex hull.png
Nonuniform
truncated icosahedron
2 5 |3
Small snub icosicosidodecahedron.png
U32
| 5/2 3 3

See also

Notes

  1. ^ Kotschick, Dieter (2006), "The Topology and Combinatorics of Soccer Balls", American Scientist 94 (4): 350–357 
  2. ^ Rhodes, Richard (1996). Dark Sun: The Making of the Hydrogen Bomb. Touchstone Books. pp. 195. ISBN 0-684-82414-0. 
  3. ^ http://www.mantleplumes.org/EarthTess.html
  4. ^ J.W. Sears, Icosahedral fracture tessellation of early Mesoproterozoic Laurentia. Geology v. 29, p. 327- 330.

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

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