At first glance, the shirt on the left looks like a soccer jock shirt – but it’s actually a very nerdy math shirt.
A soccer ball (European football) usually has 12 pentagons and 20 hexagons, arranged in a highly symmetric fashion. The polyhedron most closely resembling this familiar object is the truncated icosahedron, an Archimedean solid.
Archimedean solids are an extension of the Platonic (regular) solids. Archimedean solids have regular polygons as faces, and have symmetries which send any vertex to any other. The rotational symmetries of the truncated icosahedron are the same as those for an icosahedron or dodecahedron. There are 60 rotational symmetries, isomorphic to the alternating group A5.
The truncated icosahedron is also the shape of buckminsterfullerene, C60, the buckyball. This is a form of carbon with fascinating properties only identified in the 1980s. The discovery of fullerenes earned the 1996 Nobel Prize in chemistry. Some viruses also share this shape.
On our Truncated Icosahedron Math Shirt, we have chosen to show the Euler characteristic equation χ = vertices – edges + faces = 2. The surface of any convex polyhedron is a sphere, hence will have the Euler characteristic of a sphere, which is 2 for a 3-D polyhedron. For the truncated icosahedron, this equation is 60 – 90 + 32 = 2.
The Euler characteristic is a deep topological property connected with the Lefschetz fixed point theorem. An application to the sphere is that any orientation-preserving map from the sphere to itself must have at least two fixed points, counting multiplicity. As an example, if you rotate the sphere, there are two fixed points where the axis of rotation intersects the sphere. Another example: The complex plane plus infinity is also a sphere. The map z |-> z + 1 has only one fixed point, infinity, but this fixed point has multiplicity 2.
The Euler characteristic χ can be computed from any decomposition into polyhedra, so it is a constant for convex polyhedra. For a cube, vertices – edges + faces = 8 – 12 + 6 = 2. For a dodecahedron, vertices – edges + faces = 20 – 30 + 12 = 2. That the Euler characteristic is constant is easily guessed, but harder to prove. Elementary proofs don’t explain the importance of the Euler characteristic.
One advanced explanation for why the Euler characteristic is constant is that it is the alternating sum of the ranks of the homology groups of a space. Any map deformable to the identity, or whose action is the identity on homology, will have at least χ fixed points, counting multiplicity.
The Gauss-Bonnet theorem says the total curvature of a surface equals 2π times its Euler characteristic. A smooth sphere of radius 1 has unit curvature per area, so its total curvature is its surface area, 4π, or 2π * the Euler characteristic 2. A discrete version covers the truncated icosahedron. At each vertex, the sum of the angles of the polygons is less than 2π radians. Any polyhedral decomposition of a surface has total angular defect equal to the 2π times the Euler characteristic. If 3 polygons meet at each vertex, then the angular defect may be associated to the polygons instead of the vertices. The defect of an n-gon is 2π * (1-(n/6)), so the defect of a hexagon is 0, the defect of a pentagon is π/3, and the defect of a quadrilateral is 2π/3. Thus, both the dodecahedron and truncated icosahedron have 12 pentagons, to have total defect 4π, and both the cube and truncated octahedron have 6 squares.
Another symmetric sports ball is from sepak takraw, a popular Asian sport which may be described as kick-volleyball. The classic design for the takraw (ball) has snub-icosahedral symmetry, or icosahedral symmetry without the reflections. The hollow (whiffle-like) structure is made from reeds or plastic pieces woven around 12 pentagons in a chiral fashion, so that takraws are either right-handed or left-handed.