The Beautiful Game

What is the most beautiful rectangular shape? What is the ratio of width to height that is most aesthetically pleasing? This question has been considered by art-lovers for centuries and one value appears consistently, called the golden ratio or Divine proportion.

I must admit that the notion of an ideal ratio makes me uncomfortable. How can it possibly be demonstrated? A survey will not do: just reflect on the atrocious cacophony conjured up by a vox pop on “the most beautiful music”.

Yet the golden ratio is taken seriously by many “authorities”. A book by Luca Pacioli in 1509, The Divine Proportion, caught the imagination of many artists and architects. Enthusiasts find the ratio everywhere: in the design of the Great Pyramid, the proportions of the Parthenon, even the structure of Bartók’s music.

The golden ratio, or golden mean, was studied in ancient Greece. Technically, it emerges when we divide a line segment into two pieces such that the ratio of the longer part to the shorter equals that of the whole segment to the longer part. The ratio, often denoted by the Greek letter phi (φ), has the value (1+√5)/2 or about 1.618.

Let’s see if we can find the golden mean in the proportions of a football pitch. For international games, the length of the pitch must be between 100 and 110 metres and the width between 64 and 75 metres. So the aspect ratio, length divided by width, is somewhere between 1.333 and 1.719. The golden mean falls within this range, so perhaps we’re onto something. But the preferred dimensions are 105 by 68 metres, with a ratio of 1.554, not especially close to φ.

Measurements of a standard football pitch (image: Wikipedia Commons).

What about the penalty area? The ratio here is 2.444; we’re getting cold. What about the goal area? The ratio is now 3.331, colder still. And the most prominent rectangle, the goal itself, is eight yards wide and eight feet high, with a ratio of 3 to 1, nowhere remotely near φ.

Of course, φ-zealots or “phiophiles” will find that which they seek. By the “Strong Law of Small Numbers” there are not enough values to go round, and φ is sure to turn up sooner or later. The golden ratio does appear frequently in a number of biological systems, most notably in the arrangement of leaves on a stem and seeds in a sunflower. But, on the basis of the evidence above, we must conclude that the beautiful ratio is not apparent in the beautiful game.