Truncated icosahedron

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. Therefore, the resulting polyhedron has 32 faces, 90 edges, and 60 vertices. 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 $$ \operatorname{GP}(1,1) $$.

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: $$\begin{align} (0, \pm 1, \pm 3\varphi), \\ (\pm 1, \pm (2 + \varphi), \pm 2\varphi), \\ (\pm \varphi, \pm 2, \pm (2\varphi + 1)), \end{align}$$ 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 $$\sqrt{9\varphi + 10} \approx 4.956$$. 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: $$\begin{align} A &= \left ( 20 \cdot \frac32\sqrt{3} + 12 \cdot \frac54\sqrt{ 1 + \frac{2}{\sqrt{5}}} \right) a^2 \approx 72.607a^2 \\ V &= \frac{125+43\sqrt{5}}{4} a^3 \approx 55.288a^3. \end{align}$$

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 $$ 5 \cdot 6^2 $$. The truncated icosahedron's dual is pentakis dodecahedron, a Catalan solid, shares the same symmetry as the truncated icosahedron.

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.

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, 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.

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.

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.

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, 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.