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The Topology and Combinatorics of Soccer Balls



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Abstract:

 

Figure 4. New soccer balls...Click to Enlarge Image To a mathematician, the iconic soccer ball...Click to Enlarge Image

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.


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