Truncated icosahedron  

(Click here for rotating model) 

Type  Archimedean solid 
Elements  F = 32, E = 90, V = 60 (χ = 2) 
Faces by sides  12{5}+20{6} 
Schläfli symbol  t{3,5} 
Wythoff symbol  2 5  3 
CoxeterDynkin  
Symmetry  I_{h} or (*532) 
References  U_{25}, C_{27}, W_{9} 
Properties  Semiregular convex 
Colored faces 
5.6.6 (Vertex figure) 
Pentakis dodecahedron (dual polyhedron) 
Net 
In geometry, the truncated icosahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids whose faces are two or more types of regular polygon.
It has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges.
Contents 
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.
Icosahedron 
Cartesian 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.
The area A and the volume V of the truncated icosahedron of edge length a are:
The truncated icosahedron easily verifies the Euler characteristic:
With unit edges, the surface area is (rounded) 21 for the pentagons and 52 for the hexagons, together 73 (see areas of regular polygons).
The association football (soccer) ball is perhaps the bestknown example of a spherical polyhedron analog to the truncated icosahedron, found in everyday life. ^{[1]} 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 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. ^{[2]}
The truncated icosahedron can also be described as a model of the Buckminsterfullerene (fullerene) (C_{60}), or "buckyball," molecule, a recentlyidentified allotrope of elemental carbon. The diameter of the soccer ball and the fullerene molecule are 22 cm and ca. 1 nm, respectively, hence the size ratio is 220,000,000 : 1.
The truncated icosahedron is also hypothesized in geology to be the driving force behind many tectonic fabrics on earth. According to the theory, since the shape is the closest geometric analog to the shape of the earth, it can explain the trend of many different fracture and associated features in plate tectonic rifting and craton shape.^{[3]}^{[4]}
A truncated icosahedron with "solid edges" is a drawing by Lucas Pacioli illustrating The Divine Proportion.
These uniform starpolyhedra, and one icosahedral stellation have nonuniform truncated icosahedra convex hulls:
Nonuniform truncated icosahedron 2 5 3 
U37 2 5/2  5 
U61 5/2 3  5/3 
U67 5/3 3  2 
U73 2 5/3 (3/2 5/4) 
Complete icosahedron 

Nonuniform truncated icosahedron 2 5 3 
U38 5/2 5  2 
U44 5/3 5  3 
U56 2 3 (5/4 5/2)  

Nonuniform truncated icosahedron 2 5 3 
U32  5/2 3 3 


