Truncated icosahedron

3D view of a truncated icosahedron ( animation )
Football : projection of the surfaces of a truncated icosahedron onto the spherical surface

The truncated icosahedron (also called a football body ) is a polyhedron ( polyhedron ) that is created by blunting the corners of an icosahedron and is one of the Archimedean solids . Instead of the twelve corners of the icosahedron there are now twelve regular pentagons ; the 20 triangles of the icosahedron become regular hexagons . The polyhedron is thus composed of a total of 32 surfaces and has 60 corners and 90 edges.

In the regular truncated icosahedron, i.e. the football body, all 90 edges are of the same length.

The body that is dual to the truncated icosahedron is the pentakis dodecahedron .

By far the best-studied fullerene molecule C 60 has the structure of a truncated icosahedron.

Formulas

Sizes of a regular truncated icosahedron with edge length a
volume ${\ displaystyle V = {\ frac {a ^ {3}} {4}} \ left (125 + 43 {\ sqrt {5}} \ right)}$
Surface area ${\ displaystyle A_ {O} = 3a ^ {2} \ left (10 {\ sqrt {3}} + {\ sqrt {25 + 10 {\ sqrt {5}}}} \ right)}$
Umkugelradius ${\ displaystyle R = {\ frac {a} {4}} {\ sqrt {58 + 18 {\ sqrt {5}}}}}$
( pentagon )
${\ displaystyle \ rho _ {1} = {\ frac {a} {2}} {\ sqrt {\ frac {125 + 41 {\ sqrt {5}}} {10}}}}$
( hexagon )
${\ displaystyle \ rho _ {2} = {\ frac {a} {4}} {\ sqrt {3}} \ left (3 + {\ sqrt {5}} \ right)}$
Edge ball radius ${\ displaystyle r = {\ frac {3} {4}} \, a \ left (1 + {\ sqrt {5}} \ right)}$
1. Face angle
(hexagon – hexagon)
≈ 138 ° 11 ′ 23 ″
${\ displaystyle \ cos \, \ alpha _ {1} = - {\ frac {1} {3}} {\ sqrt {5}}}$
2. Face angle
(hexagon – pentagon)
≈ 142 ° 37 ′ 21 ″
${\ displaystyle \ cos \, \ alpha _ {2} = - {\ sqrt {\ frac {5 + 2 {\ sqrt {5}}} {15}}}}$
Corners solid angle
≈ 1.3524 π
${\ displaystyle \ Omega = 2 \ pi - \ arccos \ left (- {\ frac {1} {5}} {\ sqrt {5}} \ right)}$