Impossible Football

11 Sept

What is wrong with the pattern of white and dark hexagons stipulated in UK law as the image for a football ground in England and Wales?

20,000 people signed up petition to get the street sign changed. ( Click the link to know more Football ground street signs are wrong, says fan - BBC News)

The street sign is mathematically incorrect, as it is impossible to create a ball using only hexagons. Geometrically, football is a truncated icosahedron. This ball type was introduced to the World Cup in 1970 (this iconic design has been superseded by alternative patterns since year 2006).

Icosahedron is a 20-faced Archimedean solid with 12 vertices and 30 edges. By truncating or chopping off, each of the 12 vertices at the one-third mark of each edge, we can create 12 pentagonal faces and in the process 20 triangle faces are transformed into regular hexagons and we get truncated icosahedron, our football geometry.An outstanding H3 heading to give context.

When 12 vertices of an Icosahedron are chopped off, each of the chopped vertices create a corresponding pentagon surrounded by 5 hexagons to create truncated icosahedron.

Thus, a truncated icosahedron is a 32-faced Archimedean solid with 60 vertices corresponding to the facial arrangement of 20 (white) hexagons and 12 (black) pentagons.

**Typical Adidas Telstar football with hexagons and pentagons.**

Fun fact!
The truncated icosahedron is also known to chemists as the structure of pure carbon known as a buckyball fullerene. A fullerene is an allotrope of carbon whose molecule consists of carbon atoms denoted by their empirical formula Cn, often written Cn, where n is the number of carbon atoms. Buckminsterfullerene is a type of fullerene with the formula C60. It has a cage-like fused-ring structure (truncated icosahedron) made of twenty hexagons and twelve pentagons, and resembles a football. Each of its 60 carbon atoms is bonded to its three neighbours.