The Topology and Combinatorics of Soccer Balls

W the arrival of the quadrennial soccer World Cup this summer, more than a billion people around the world are finding their television and computer screens filled with depictions of soccer halls. In Germany, where the World Cup matches are being played, soccer balls are tuming up on all kinds of merchandise, much of it having nothing to do with soccer. Although a soccer ball can be put together in many different ways, there is one design so ubiquitous that it has become iconic. This standard soccer ball is stitched or glued together from 32 polygons, 12 of them five-sided and 20 sixsided, arranged in such a way that every pentagon is surrounded hy hexagons. Postmodern paint jobs notwithstandiiig, the traditional way to color such a ball is to paint the pentagons black and the hexagons white. This color scheme was reportedly introduced for the World Cup in 1970 to enhance the visibility of the ball on television, although the design itself is older. Most people associate the soccer-ball image with hours spent on the field or the sidelines, or perhaps just with advertisements for sport merchandise. But to a mathematician, a soccer bail is an intriguing puzzle. Why does it look the way it does? Are there other ways of putting it together? Could the penta-