Everything about The Truncated Icosahedron totally explained
The
truncated icosahedron is an
Archimedean solid. It comprises 12 regular
pentagonal faces, 20 regular
hexagonal faces, 60 vertices and 90 edges.
Construction
This polyhedron can be constructed from an
icosahedron with the 12 vertices
truncated (cut off) such that one third of each edge is cut off at each of both ends. This creates 12 new pentagon faces, and leaves the original 20 triangle faces as regular hexagons. Thus the length of the edges is one third of that of the original edges.
Canonical coordinates
Canonical coordinates for the vertices of a
truncated icosahedron centered at the origin are the
orthogonal rectangles (0,±1,±3φ), (±1,±3φ,0), (±3φ,0,±1) and the orthogonal
cuboids (±2,±(1+2φ),±φ), (±(1+2φ),±φ,±2), (±φ,±2,±(1+2φ)) along with the orthogonal cuboids (±1,±(2+φ),±2φ), (±(2+φ),±2φ,±1), (±2φ,±1,±(2+φ)), where φ = (1+√5)/2 is the
golden mean. Using φ
2 = φ + 1 one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to 9φ + 10. The edges have length 2.
Area and volume
The area
A and the volume
V of the truncated icosahedron of edge length
a are:
»
Geometric relations
The truncated icosahedron easily verifies the
Euler characteristic:
» 32 + 60 − 90 = 2.
With unit edges, the surface area is (rounded) 21 for the pentagons and 52 for the hexagons, together 73 (see
areas of regular polygons).
Applications
A
soccer ball is an 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 is more spherical due to the pressure of the air inside and the elasticity of the ball.
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 (C
60) molecule. The diameter of the soccer ball and the Buckminsterfullerene molecule are 22 cm and ca. 1
nm, respectively, hence the size ratio is 220,000,000 : 1.
Truncated icosahedra in the arts
A truncated icosahedron with "solid edges" is a drawing by
Lucas Pacioli illustrating
The Divine Proportion.
Further Information
Get more info on 'Truncated Icosahedron'.
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