The Topology and Combinatorics of Soccer Balls
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Abstract:
The standard soccer ball, a spherical polyhedron made up of 12
(traditionally black) pentagons and 20 (traditionally white)
hexagons, is an object of more than sporting interest. It shares its
geometry with the carbon-60 molecule or "buckyball" and
has inspired considerable work in group theory. But mathematicians
can design many more soccer balls, extending the basic design using
the tools of topology and combinatorics. Using a construction called
a "branched covering," a topologist can slice, duplicate
and reattach the standard ball's cover, even creating toroidal
soccer balls. Generalizing the rules of soccer ball opens up further
possibilities and mathematical questions for exploration.