A beautiful number: the golden ratio

Some numbers are held to be lucky, others unlucky and yet others are imbued with divine or mystical significance.

One, however, is associated with beauty. Known as the golden ratio, its repeat appearance in architecture, design, nature and the proportions of the human body has long been a source of fascination to mathematicians and artists alike.

In mathematics, two numbers are said to be in a golden ratio if the ratio of the larger (a) to the smaller (b) is the same as the ratio of their sum (a +b) to the larger number (a). That is, (a + b)/a = a/b. Expressed diagrammatically.

The value of the golden ratio is ½(1 + √5), an irrational number: 1.6180889887 …

The Greek sculptor Phidias is said to have made use of the golden ratio in his design of sculptures in the Parthenon (none of which survive, unfortunately) and the first letter of his name, the Greek letter phi φ is used as the symbol for the ratio.

Leonardo Da Vinci used the golden ratio in his illustrations of geometric figures in Lucia Pacioli’s manuscript De Divina Proportione (On the Divine Proportion), an influential treatise on architecture and the human body. The golden ratio is also evident in early Islamic architecture and in Gothic cathedrals such as Notre Dame in Paris and Chartres. Many books produced between 1550 and 1770 use this exact proportion.

Today, the golden ratio crops up in contemporary tessellation patterns. Any floor tiler will tell you they can use triangles, squares and hexagons to create a pattern with a three, four or six-fold symmetry. What about five-fold symmetry? It is impossible to tessellate a plane using pentagons, but in the 1970s the Oxford mathematician Roger Penrose proved that a perfect tessellation pattern with five-fold symmetry could be created using a combination of two tiles shaped like a fat and a thin rhombus.

The ratio of the sides to the long diagonal of the fat rhombus turns out to be – you’ve guessed it – the golden ratio, φ. While for the thin rhombus, the ratio of the sides to the short diagonal is 1/φ. The Penrose tiling pattern appears in jigsaws, textbooks and, fittingly, on the patio of the Mathematical Institute at Oxford University. Some years after Penrose published his unusual pattern, scientists were astonished to discover crystals displaying a five-fold symmetry, previously thought impossible (they are known as quasi-crystals because they do not have a periodically repeating structure).

I have a curious personal association with φ. Forty years ago Stephen Hawking announced that black holes are not black but glow with heat radiation. We are familiar with hot bodies that cool down as they emit heat (think of your abandoned coffee going cold). Black holes do the opposite – they get hotter. In technical language black holes are said to have a negative specific heat. This means a black hole is unstable: the more it radiates, the hotter it gets and the faster it emits heat. This runaway process results in the black hole evaporating at an escalating rate before disappearing in an explosion.

I was at the lecture in which Hawking announced this astonishing result, and for a while I found it hard to believe. I began to take a close interest in the thermodynamic properties of black holes. I used a mathematical model to see how things might change if the black hole is spinning. To my surprise, I found that when the black hole spins fast enough, its specific heat is positive; that is, it cools as it radiates heat – like that cup of coffee. Now I should explain that there is a maximum spin rate for a black hole of a given mass, above which it would cease to be a black hole and turn into a so-called naked singularity – something that many physicists believe is impossible. Intriguingly, the flip-over between negative and positive specific heat occurs when the square of the spin rate reaches 1/φ of the maximum.

I discovered this peculiar fact in 1979 and still don’t know what to make of it. As far as can be seen, there is no dramatic alteration to the structure or shape of the black hole at the transition point. Perhaps there is something in the warped architecture of the black hole that picks out φ, just as Phidias reputedly did with his Parthenon sculptures. Or perhaps the golden ratio is more deeply embedded in the structure of nature and this is one glimpse of it.

Whatever the case, this example – and others that have fascinated mathematicians and artists for centuries – illustrates that what is significant in nature is also likely to be beautiful.