Now that the World Cup is over, it's time for me to tell you that a soccer ball doesn't need to be spherical. With the help of Ivars Peterson in his last Math Trek column, "Bending a Soccer Ball," let's see how the FIFA soccer balls used between 1970 and 2006 could be modified and "improved." The former balls were composed of 32 pieces of material, 12 pentagons and 20 hexagons, assembled according to very strict rules. But some mathematicians have relaxed some of these rules and have modeled soccer balls looking like doughnuts. In the mean time, German designers have built a collection of 32 soccer balls, one for each team who participated to the competition won by Italy over France.
Here is an introduction from this Math Trek column, published by Science News Online.
Mathematics, however, suggests that the older, 32-panel design can itself be modified in various ways to yield a variety of new patterns for spherical and even doughnut-shaped soccer balls.
"To a mathematician, a soccer ball is an intriguing puzzle," mathematician Dieter Kotschick of the University of Munich writes in the July-August [2006 issue of] American Scientist.
Here are two links to the abstract of this paper, "The Topology and Combinatorics of Soccer Balls" and to another paper by Kotschick and one of his colleagues, "The classification of football patterns" (PDF format, 13 pages, 142 KB) which starts like this.
A football pattern is a graph embedded in the two-sphere in such a way that all faces are pentagons and hexagons, satisfying the conditions that the edges of each pentagon meet only edges of hexagons, and that the edges of each hexagon alternately meet edges of pentagons and of hexagons. If one requires that there are exactly three faces meeting at each vertex, then Euler’s formula implies that the pattern consists of 12 pentagons and 20 hexagons.
But if one requirement is dropped, Peterson says that it would be "possible to create new soccer balls by using a mathematical construction called a branched covering."
You'll find explanations about branched covering in plain English in Peterson's article, and in mathematical terms in Kotschick's paper, which was illustrated by images calculated by Michael Trott of Wolfram Research and animated by Amy Young.
You can see these images and animations on Michael Trott's web site by reading "Bending a Soccer Ball -- Mathematically, by Michael Trott" (June 2006). For example, below is a morphing of a torus into a double-covered soccer ball (Credit: Michael Trott, Wolfram Research).
Here is a link to a short movie in QuickTime format showing "how a torus is continuously deformed into two concentric soccer balls of identical size and orientation. No tearing of the surface occurs in this transition. The typically black pentagons of a soccer ball are colored differently for easy visual tracking of the morphing. The two final, staggered soccer balls are connected at four points at vertices of four of the twelve black pentagons."
But a soccer ball can still be spherical and describe the country it represents. This was a theme of an image collection designed by bora.herke from Germany. This company has designed 32 qualified balls, one for each country represented at the World Cup. These soccer balls were on display in Paris at Colette between May and June and in an Adidas store in Berlin until today.
Each soccer ball is covered by a product symbolizing the country, such as denim for the U.S. or tabloids for England. Below are the two balls representing France and Italy, the two finalists of the World Cup 2006 (Credit: bora.herke).
And who won? Italy, after 120 minutes ending with a 1-1 score and a penalty shootout won 5-3.
Sources: Ivars Peterson, Math Trek, Science News Online, July 8, 2006; and various web sites
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