Archimedean solids which are semi-regular. Examples can be generated from the
Platonic solids. If we slice off (truncate) some corners of the icosahedron we
have the shape used as the design for the modern soccer ball. The 32 faces that
form the panels are made up of 12 pentagons and 20 hexagons. There are 90
edges and 60 vertices. It is also the shape of buckminsterfullerene molecules,
named after the visionary Richard Buckminster Fuller, creator of the geodesic
dome. These ‘bucky balls’ are a newly discovered form of carbon, C 60 , with a
carbon atom found at each vertex.
Euler’s formula
Euler’s formula is that the number of vertices V, edges E and faces F, of a
polyhedron are connected by the formula
V – E + F = 2
For example, for a cube, V = 8, E = 12 and F = 6 so V – E + F = 8 – 12 + 6
= 2 and, for buckminsterfullerene, V – E + F = 60 – 90 + 32 = 2. This theorem
actually challenges the very notion of a polyhedron.
The cube with a tunnel
If a cube has a ‘tunnel’ through it, is it a real polyhedron? For this shape, V =
16, E = 32, F = 16 and V – E + F = 16 – 32 + 16 = 0. Euler’s formula does not
work. To reclaim the correctness of the formula, the type of polyhedron could be
limited to those without tunnels. Alternatively, the formula could be generalized
to include this peculiarity.