Truncated icosahedron

Properties

Metric properties

The surface area ${\displaystyle A}$ and the volume ${\displaystyle V}$ of the truncated icosahedron of edge length ${\displaystyle a}$ are: $${\displaystyle \begin{array}{cc}A& =\left(20\cdot \frac{3}{2}\sqrt{3}+12\cdot \frac{5}{4}\sqrt{1+\frac{2}{\sqrt{5}}}\right)a^2\approx 72.607a^2\\ V& =\frac{125+43\sqrt{5}}{4}{a}^3\approx 55.288a^3.\end{array}}$$ The sphericity of a polyhedron ${\displaystyle \mathrm{\Psi }}$ describes how closely a polyhedron resembles a sphere. It can be defined as the ratio of the surface area of a sphere with the same volume to the polyhedron's surface area, from which the value is between 0 and 1. In the case of a truncated icosahedron, it is: $${\displaystyle \mathrm{\Psi }=\frac{6{\pi }^{1/2}V}{A^{3/2}}\approx \mathrm{0.9504.}}$$

The dihedral angle of a truncated icosahedron between adjacent hexagonal faces is approximately 138.18°, and that between pentagon-to-hexagon is approximately 142.6°.

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. It has the same symmetry as the regular icosahedron, the icosahedral symmetry, and it also has the property of vertex-transitivity. The polygonal faces that meet for every vertex are one pentagon and two hexagons, and the vertex figure of a truncated icosahedron is ${\displaystyle 5\cdot 6^2}$. The truncated icosahedron's dual is pentakis dodecahedron, a Catalan solid, shares the same symmetry as the truncated icosahedron.

The truncated icosahedron has a Rupert property, meaning that for the same size or a smaller size of it can pass through a hole.

Graph

According to Steinitz's theorem, the skeleton of a truncated icosahedron, like that of any convex polyhedron, can be represented as a polyhedral graph, meaning a planar graph (one that can be drawn without crossing edges) and 3-vertex-connected graph (remaining connected whenever two of its vertices are removed). The graph is known as truncated icosahedral graph, and 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 vertex is incident to exactly three edges.

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. The ball comprises the same pattern of regular pentagons and regular hexagons, each of which is painted in black and white, respectively; still, its shape is more spherical. It was introduced by Adidas, which debuted the Telstar ball during the World Cup in 1970. However, it was superseded in 2006.

Geodesic domes are typically based on triangular facetings of this geometry, with example structures found across the world, popularized by Buckminster Fuller. An example can be found in the model of a buckminsterfullerene or buckyball, a truncated icosahedron-shaped geodesic dome allotrope of elemental carbon discovered in 1985. In other engineering and science applications, its 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. Its structure can also be found in the protein of clathrin.

The truncated icosahedron was known to Archimedes, who classified the 13 Archimedean solids in a lost work. All that is now known 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 are from a rediscovery by Piero della Francesca, in his 15th-century book De quinque corporibus regularibus, 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.

Related polyhedra

The truncated icosahedron, which has 32 faces (12 pentagons and 20 hexagons) looks similar to the uniform chamfered dodecahedron (truncated rhombic triacontahedron), which has 42 faces (12 pentagons and 30 hexagons).