Truncated icosahedron

Truncated icosahedron
Truncated icosahedron
Type	Archimedean solid; Uniform polyhedron; Goldberg polyhedron
Faces	32
Edges	90
Vertices	60
Symmetry group	Icosahedral symmetry
Dual polyhedron	Pentakis dodecahedron
Vertex figure
Net

In geometry, the truncated icosahedron is an Archimedean solid with 32 faces. It is a polyhedron that may be associated with footballs (soccer balls) typically patterned with white hexagons and black pentagons; the Adidas Telstar was the first soccer ball to use this pattern in the 1970s. 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 an example of a Goldberg polyhedron.

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

Construction

The truncated icosahedron can be constructed from a regular icosahedron by cutting off all of its vertices, known as truncation. Each of the 12 vertices at the one-third mark of each edge creates 12 pentagonal faces and transforms the original 20 triangle faces into regular hexagons.^[1]^[2] Therefore, the resulting polyhedron has 32 faces, 90 edges, and 60 vertices.^[3] A Goldberg polyhedron is one whose faces are 12 pentagons and some multiple of 10 hexagons. There are three classes of Goldberg polyhedron, one of them is constructed by truncating all vertices repeatedly, and the truncated icosahedron is one of them, denoted as $\operatorname {GP} (1,1)$ .^[4]

One way to construct a truncated icosahedron with edge length 2, centered at the origin, is with Cartesian coordinates that are all even permutations of: ${\begin{aligned}(0,\pm 1,\pm 3\varphi ),\\(\pm 1,\pm (2+\varphi ),\pm 2\varphi ),\\(\pm \varphi ,\pm 2,\pm (2\varphi +1)),\end{aligned}}$ where $\varphi ={\tfrac {1+{\sqrt {5}}}{2}}$ is the golden mean.

Properties

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: $r_{\mathrm {u} }={\frac {a}{2}}{\sqrt {1+9\varphi ^{2}}}={\frac {a}{4}}{\sqrt {58+18{\sqrt {5}}}}\approx 2.478a$ where $φ$ is the golden ratio. The circumradius is ${\sqrt {9\varphi +10}}\approx 4.956$ .^[5] 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 a shared edge (calculated based on this construction) is approximately 23.281446°.

The surface area $A$ and the volume $V$ of the truncated icosahedron of edge length $a$ are:^[3] ${\begin{aligned}A&=\left(20\cdot {\frac {3}{2}}{\sqrt {3}}+12\cdot {\frac {5}{4}}{\sqrt {1+{\frac {2}{\sqrt {5}}}}}\right)a^{2}\approx 72.607a^{2}\\V&={\frac {125+43{\sqrt {5}}}{4}}a^{3}\approx 55.288a^{3}.\end{aligned}}$

The truncated icosahedron is an Archimedean solid, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex.^[6] It has the same symmetry as the regular icosahedron, the icosahedral symmetry, and it also has the property of vertex-transitivity.^[7]^[8] The polygonal faces that meet for every vertex are one pentagon and two hexagons, and the vertex figure of a truncated icosahedron is $5\cdot 6^{2}$ . The truncated icosahedron's dual is pentakis dodecahedron, a Catalan solid,^[9] shares the same symmetry as the truncated icosahedron.^[10]

Truncated icosahedral graph

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. It is an Archimedean graph, because it resembles one of the Archimedean solids. It is a cubic graph, meaning that each of its vertices is connected by the other three vertices.^[11]^[12]^[13]

Appearance

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.^[14] 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.^[15]

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.^[16]

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.

Gallery
Truncated icosahedral radome on a weather station
Truncated icosahedron machined out of 6061-T6 aluminum
A wooden truncated icosahedron artwork by George W. Hart.

History

The truncated icosahedron was known to Archimedes, who classified the 13 Archimedean solids in a lost work. However, 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,^[17] 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.^[18]

References

^ Mednikov, Evgueni G.; Jewell, Matthew C.; Dahl, Lawrence F. (2007-09-01). "Nanosized (μ 12 -Pt)Pd 164- x Pt x (CO) 72 (PPh 3 ) 20 ( x ≈ 7) Containing Pt-Centered Four-Shell 165-Atom Pd−Pt Core with Unprecedented Intershell Bridging Carbonyl Ligands: Comparative Analysis of Icosahedral Shell-Growth Patterns with Geometrically Related Pd 145 (CO) x (PEt 3 ) 30 ( x ≈ 60) Containing Capped Three-Shell Pd 145 Core". Journal of the American Chemical Society. 129 (37): 11624. doi:10.1021/ja073945q. ISSN 0002-7863. PMID 17722929.
^ Kotschick, Dieter (July–August 2006). "The Topology and Combinatorics of Soccer Balls". American Scientist. 94 (4): 350. doi:10.1511/2006.60.350.
^ ^a ^b Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.
^ Hart, George (2012). "Goldberg Polyhedra". In Senechal, Marjorie (ed.). Shaping Space (2nd ed.). Springer. pp. 125–138. doi:10.1007/978-0-387-92714-5_9. ISBN 978-0-387-92713-8.
^ Weisstein, Eric W. "Icosahedral group". MathWorld.
^ Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. Springer. p. 39. doi:10.1007/978-3-319-64123-2. ISBN 978-3-319-64123-2.
^ Koca, M.; Koca, N. O. (2013). "Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes". Mathematical Physics: Proceedings of the 13th Regional Conference, Antalya, Turkey, 27–31 October 2010. World Scientific. p. 48.
^ Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. p. 386. ISBN 978-0-521-55432-9.
^ Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 90.
^ Holden, Alan (1991). Shapes, Space, and Symmetry. Dover Books on Mathematics. Courier Corporation. p. 52. ISBN 9780486268514.
^ 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
^ Kotschick, Dieter (2006). "The Topology and Combinatorics of Soccer Balls". American Scientist. 94 (4): 350–357. doi:10.1511/2006.60.350.
^ 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.
^ Rhodes, Richard (1996). Dark Sun: The Making of the Hydrogen Bomb. Touchstone Books. pp. 195. ISBN 0-684-82414-0.
^ 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.
^ Field, J. V. (1997). "Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler". Archive for History of Exact Sciences. 50 (3–4): 241–289. doi:10.1007/BF00374595. JSTOR 41134110. MR 1457069. S2CID 118516740.

External links

Weisstein, Eric W., "Truncated icosahedron" ("Archimedean solid") at MathWorld.
- Weisstein, Eric W. "Truncated icosahedral graph". 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

[mjd-1] Mednikov, Evgueni G.; Jewell, Matthew C.; Dahl, Lawrence F. (2007-09-01). "Nanosized (μ 12 -Pt)Pd 164- x Pt x (CO) 72 (PPh 3 ) 20 ( x ≈ 7) Containing Pt-Centered Four-Shell 165-Atom Pd−Pt Core with Unprecedented Intershell Bridging Carbonyl Ligands: Comparative Analysis of Icosahedral Shell-Growth Patterns with Geometrically Related Pd 145 (CO) x (PEt 3 ) 30 ( x ≈ 60) Containing Capped Three-Shell Pd 145 Core". Journal of the American Chemical Society. 129 (37): 11624. doi:10.1021/ja073945q. ISSN 0002-7863. PMID 17722929.

[kotschick-2] Kotschick, Dieter (July–August 2006). "The Topology and Combinatorics of Soccer Balls". American Scientist. 94 (4): 350. doi:10.1511/2006.60.350.

[berman-3] Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.

[hart-4] Hart, George (2012). "Goldberg Polyhedra". In Senechal, Marjorie (ed.). Shaping Space (2nd ed.). Springer. pp. 125–138. doi:10.1007/978-0-387-92714-5_9. ISBN 978-0-387-92713-8.

[5] Weisstein, Eric W. "Icosahedral group". MathWorld.

[diudea-6] Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. Springer. p. 39. doi:10.1007/978-3-319-64123-2. ISBN 978-3-319-64123-2.

[kocakoca-7] Koca, M.; Koca, N. O. (2013). "Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes". Mathematical Physics: Proceedings of the 13th Regional Conference, Antalya, Turkey, 27–31 October 2010. World Scientific. p. 48.

[cromwell-8] Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. p. 386. ISBN 978-0-521-55432-9.

[williams-9] Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 90.

[holden-10] Holden, Alan (1991). Shapes, Space, and Symmetry. Dover Books on Mathematics. Courier Corporation. p. 52. ISBN 9780486268514.

[11] Read, R. C.; Wilson, R. J. (1998). An Atlas of Graphs. Oxford University Press. p. 268.

[12] Godsil, C. and Royle, G. Algebraic Graph Theory New York: Springer-Verlag, p. 211, 2001

[13] Kostant, B. The Graph of the Truncated Icosahedron and the Last Letter of Galois. Notices Amer. Math. Soc. 42, 1995, pp. 959-968 PDF

[14] Kotschick, Dieter (2006). "The Topology and Combinatorics of Soccer Balls". American Scientist. 94 (4): 350–357. doi:10.1511/2006.60.350.

[15] 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.

[16] Rhodes, Richard (1996). Dark Sun: The Making of the Hydrogen Bomb. Touchstone Books. pp. 195. ISBN 0-684-82414-0.

[17] 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.

[18] Field, J. V. (1997). "Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler". Archive for History of Exact Sciences. 50 (3–4): 241–289. doi:10.1007/BF00374595. JSTOR 41134110. MR 1457069. S2CID 118516740.

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