Why does a football always have exactly 12 black pentagons?

Think of a honeycomb. When three regular hexagons meet at a corner, their angles add up to exactly 360 degrees, so they form a flat floor with no gaps. No matter how many you add, the floor never bends.

Each angle of a pentagon is 108 degrees, so when one pentagon and two hexagons meet, the total falls a little short of 360 degrees. That shortfall makes the surface fold inward, creating a curve. In effect, the pentagons are what roll the ball into a round shape.

If three pieces meet at every corner and the pieces are only pentagons and hexagons, a closed ball always needs exactly 12 pentagons. It doesn't matter whether there are 20 hexagons or 30. The answer follows directly from the relationship between the numbers of corners, edges and faces worked out by the 18th-century mathematician Euler.

So even if you add more hexagons to make a bigger ball, there are always 12 pentagons. The football shape is what you get by cutting off the 12 corners of an icosahedron, a solid with 20 triangular faces, so you can think of each cut as leaving one pentagon behind.

This 32-piece ball became known around the world around the time of the Telstar, the official ball of the 1970 World Cup in Mexico. Many households watched matches on black-and-white television then, and a ball mixing black pentagons and white hexagons showed up well on screen, with its spin easy to see.

Still, once the leather pieces are stitched together and the ball is inflated, each piece bulges outward, so a real ball is much closer to a sphere than to a polyhedron.

Since 2006, official World Cup balls have used far fewer pieces. The 2006 ball had 14 pieces and the 2014 ball had 6, and some are joined with heat instead of stitching. Fewer pieces and shorter seams make it easier to make a ball evenly round.

The same shape also appears in science. A molecule of 60 carbon atoms discovered in 1985 has exactly the shape of a football and is called the "buckyball"; its discovery won the 1996 Nobel Prize in Chemistry.