## The Football

The familiar panelled football consists of 20 identical hexagons and 12 identical pentagons. Its mathematical name is "truncated icosahedron".A hexagon drawn flat on the ground has internal angles all equal to 120 degrees, and a pentagon flat on the ground has internal angles all equal to 108 degrees. Because the internal angles of a hexagon divide 360 (360/120=3) we can cover the plane with regular hexagons leaving no gaps. In other words we can tessellate the plane. We cannot tessellate the plane with pentagons because the internal angle of a regular flat pentagon is 108 degrees, and 360/108 is not an integer.

We cannot cover a football with identical hexagons because a football is not flat. At each point on a flat surface the angle at the centre of a surface is 360. The angle at each point on a football is less than this. We can take the point on a football where two hexagons and a pentagon meet. The angle is 2 x 120+108=348 degrees.