When Math Geeks Think About Soccer

Have you ever thought about why most soccer balls look the same in terms of the number and shape of the panels? The Teamgeist ball introduced for the 2006 World Cup had a revolutionary panel design (a­nd a revolutionary price!) and adidas continues to come up with neat panel designs including their F50 Xite ball. But a soccer ball is all about geometry. An ar­ticle in Science News looks into the geometry of both old and new soccer ball designs:

This soccer ball is modeled on a geometric shape called a truncated icosahedron, which has 60 vertices. Its polygonal faces, however, are curved rather than flat so that the ball is roughly spherical.

Since about 1970, soccer balls have been stitched or glued together from 32 patches of material-12 of them five-sided and 20 of them six-sided. These patches are arranged so that each pentagon is surrounded by hexagons.

You learn something new every day!

The best part is when they decide to try out some new computer modeled designs:

“To a mathematician, a soccer ball is an intriguing puzzle,” mathematician Dieter Kotschick of the University of Munich writes in the July-August American Scientist.

­”Why does it look the way it does?” he asks. “Are there other ways of putting it together? Could the pentagons and hexagons be arranged differently? Could other polygons be used instead of pentagons and hexagons?”

Kotschick notes that, from a graph theory viewpoint, the standard soccer ball meets three important requirements.

  1. It is a polyhedron the consists of only pentagons and hexagons.
  2. The sides of each pentagon meet only hexagons.
  3. The sides of each hexagon alternately meet pentagons and hexagons.

If you require that there are exactly three faces meeting at each vertex, then the pattern must consist of 12 pentagons and 20 hexagons. If you lift this requirement, many other designs become possible.

The results are interesting to say the least. They link to another site that has a number of graphical animations of soccer balls morphing into different shapes. And if you ever wondered if a soccer ball could be formed out of one flat piece of paper cut into a certain shape, you won’t be disappointed. It can.


I’m not sure a doughnut shaped soccer ball would work too well though! :)

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