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 shared edge (calculated on the basis of this construction) is approximately 23.281446°. Area and volume The area A and the volume V of the truncated icosahedron of edge length a are: 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 truncated icosahedron easily demonstrates the Euler characteristic: 32 + 60 − 90 = 2. Applications 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 This ball type was introduced to the World Cup in 1970 (starting in 2006, this iconic design has been superseded by alternative patterns). British traffic signs indicating football grounds use a uniformly-colored hexagonal tiling section to represent a football, rather than a truncated icosahedron. This angered mathematician and comedian Matt Parker, who started a petition to the UK government to have these signs changed to be geometrically accurate. The petition was ultimately declined. Geodesic domes are typically based on triangular facetings of this geometry with example structures found across the world, popularized by Buckminster Fuller. A variation of the icosahedron was used as the basis of the honeycomb wheels (made from a polycast material) used by the Pontiac Motor Division between 1971 and 1976 on its Trans Am and Grand Prix. 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. In the arts Related polyhedra These uniform star-polyhedra, and