Truncated icosahedron
Truncated icosahedron | |
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(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} |
t0,1{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
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
Construction
This polyhedron can be constructed from an icosahedron by truncating, or cutting off, each of the 12 vertices at the one-third mark of each edge, creating 12 pentagonal faces and transforming the original 20 triangle faces into regular hexagons.[1][2]
Characteristics
In geometry and graph theory, there are some standard polyhedron characteristics.
Cartesian coordinates
where is the golden mean. The circumradius is and the edges have length 2.[3]
Orthogonal projections
The truncated icosahedron has five special
Centered by | Vertex | Edge 5-6 |
Edge 6-6 |
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
pentagon-centered |
hexagon-centered | |
Orthographic projection | Stereographic projections |
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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[4] are:
Applications
The balls used in
Geodesic domes are typically based on triangular facetings of this geometry with example structures found across the world, popularized by Buckminster Fuller.[6]
This shape was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in both
The truncated icosahedron can also be described as a model of the
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.
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Fullerene C60 molecule
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Truncated icosahedral radome on a weather station
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Truncated icosahedron machined out of6061-T6 aluminum
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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 | |
n-kis 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 star polyhedra with truncated icosahedra convex hulls | ||||||||||||
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|
This polyhedron looks similar to the uniform chamfered dodecahedron which has 12 pentagons, but 30 hexagons.
Truncated icosahedral graph
Truncated icosahedral graph | |
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Hamiltonian, regular, zero-symmetric | |
Table of graphs and parameters |
In the
5-fold symmetry |
5-fold 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 15th-century book De quinque corporibus regularibus,[11] 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.[12]
See also
- Fullerene
- Buckminsterfullerene (C60)
- Hyperbolic soccerball
- Snyder equal-area projection
- Soccer ball
Notes
- PMID 17722929.
- .
- ^ Weisstein, Eric W. "Icosahedral group". MathWorld.
- ^ Weisstein, Eric W. "Truncated Icosahedron". mathworld.wolfram.com. Retrieved 2023-09-10.
- .
- ^ Krebs, Albin (July 2, 1983). "R. Buckminster Fuller Dead; Futurist Built Geodesic Dome". The New York Times. New York, N.Y. p. 1. Retrieved 7 November 2021.
- ISBN 0-684-82414-0.
- ^ Read, R. C.; Wilson, R. J. (1998). An Atlas of Graphs. Oxford University Press. p. 268.
- ^ Godsil, C. and Royle, G. Algebraic Graph Theory New York: Springer-Verlag, p. 211, 2001
- ^ Kostant, B. The Graph of the Truncated Icosahedron and the Last Letter of Galois. Notices Amer. Math. Soc. 42, 1995, pp. 959-968 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.
- S2CID 118516740.
References
- ISBN 0-486-23729-X. (Section 3-9)
- Cromwell, P. (1997). "Archimedean solids". Polyhedra: "One of the Most Charming Chapters of Geometry". Cambridge: Cambridge University Press. pp. 79–86. OCLC 180091468.
External links
- Weisstein, Eric W., "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