In geometry, the truncated icosahedron is a polyhedron that can be constructed by truncating all of the regular icosahedron's vertices. The polyhedron can be regarded as a football (UK English) or a soccer ball (US English), typically patterned with white hexagons and black pentagons. Geodesic dome structures, such as those whose architecture Buckminster Fuller pioneered, are often based on this structure. It is an example of an Archimedean solid, as well as a Goldberg polyhedron.
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 triangular faces into regular hexagons. Therefore, the resulting polyhedron has 32 faces (12 pentagons and 20 hexagons), 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 polyhedra, one of which is constructed by repeatedly truncating all of its vertices, and the truncated icosahedron is one of them, denoted as GP ( 1 , 1 ) {\displaystyle \operatorname {GP} (1,1)} .
Properties
Metric properties The surface area A {\displaystyle A} and the volume V {\displaystyle V} of the truncated icosahedron of edge length a {\displaystyle a} are:
A = ( 20 ⋅ 3 2 3 + 12 ⋅ 5 4 1 + 2 5 ) a 2 ≈ 72.607 a 2 V = 125 + 43 5 4 a 3 ≈ 55.288 a 3 . {\displaystyle {\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 sphericity of a polyhedron Ψ {\displaystyle \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:
Ψ = 6 π 1 / 2 V A 3 / 2 ≈ 0.9504. {\displaystyle \Psi ={\frac {6\pi ^{1/2}V}{A^{3/2}}}\approx 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 5 ⋅ 6 2 {\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
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