Truncated icosahedron


Truncated icosahedron

The truncated icosahedron is an Archimedean solid. It comprises 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.

Canonical coordinates

Canonical 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::egin{align}A & = 3 left ( 10sqrt{3} + sqrt{5} sqrt{5 + 2sqrt{5 ight ) a^2 approx 72.607253a^2 \V & = frac{1}{4} (125+43sqrt{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

A soccer ball is an example of a spherical polyhedron analog to the truncated icosahedron, found in everyday life. [ Citation | title=The Topology and Combinatorics of Soccer Balls | first1=Dieter | last1=Kotschick | journal=American Scientist | year=2006 | volume=94 | issue=4 | pages=pp. 350-357 ] The ball comprises the same pattern of regular pentagons and regular hexagons, but 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. [cite book
author=Rhodes, Richard
title=Dark Sun: The Making of the Hydrogen Bomb
isbn=0-684-82414-0
publisher=Touchstone Books
year=1996
pages=195
]

The truncated icosahedron can also be described as a model of the Buckminsterfullerene (C60) molecule. The diameter of the soccer ball and the Buckminsterfullerene molecule are 22 cm and ca. 1 nm, respectively, hence the size ratio is 220,000,000 : 1.

Truncated icosahedra in the arts

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

ee also

*
*dodecahedron
*icosidodecahedron
*truncated dodecahedron
*truncated rhombic triacontahedron
*hyperbolic soccerball
*fullerene

References

Further reading

* (Section 3-9)

External links

*
* [http://www.software3d.com/TruncIcos.php Paper model of the truncated icosahedron]
* [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
* [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
* [http://www.korthalsaltes.com/ Paper Models of Polyhedra] Includes a paper model(net) of a Truncated Icosahedron
* [http://www.sw3d.net/sw3d/display.php?model=old+swf39 Bouncing Truncated Icosahedron] Virtual Reality (needs JVM)
* [http://www.lifeisastoryproblem.org/explore/net_truncated_icosahedron.pdf Printable Net of a Truncated Icosahedron] [http://www.lifeisastoryproblem.org Life is a Story Problem.org]


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