The golden ratio, without the legends.

Take a line and cut it so that the whole line is to the long piece as the long piece is to the short one. The cut lands a little past six tenths of the way along, and the long piece is about 1.618 times the short one. That number is the golden ratio.

The first person we know for certain wrote it down was Euclid, in his geometry book the Elements¹, around 300 BC. He did not call it golden. He called it cutting a line in “extreme and mean ratio”, and he needed it for a practical reason: you cannot draw a perfect five-sided figure without it. In a regular pentagon, each diagonal is 1.618 times as long as a side, and the five-pointed star inside cuts itself in the same ratio at every crossing.

You will often read that the followers of Pythagoras discovered it and kept the star as their secret sign. That may be true, but the evidence comes from writers who lived centuries later, and historians² still argue about it.

In 1498 a friar named Luca Pacioli finished a book about it, printed in 1509 as On the Divine Proportion. His friend Leonardo da Vinci drew its pictures of solid shapes, the only book Leonardo ever illustrated. Yet Pacioli advised painters and architects to use simple ratios, not this one. A century later the astronomer Johannes Kepler noticed that 5 is to 8 very nearly as 8 is to 13, and 13 to 21, and wondered if this was how plants grew. He wrote it in a little book about snowflakes that he gave a friend as a New Year’s present.

For two thousand years this was a useful piece of geometry. Nobody called it golden, and nobody said it was beautiful.

The word “golden” turns up in a German book as early as 1717, and only became common in the 1800s. Then, in 1854, a German professor named Adolf Zeising measured a great many human bodies and announced that the golden section was a law running through all of nature and art. Almost every claim you have heard about beauty and the golden ratio goes back to him.

In 1876 the scientist Gustav Fechner put it to a test. He laid out ten rectangles and asked people to choose their favourite. The golden one won, but only just: three in four choices were spread across it and its two neighbours. Researchers who have tried since have mostly found the preference weak³. Around 1909 an American engineer, Mark Barr, gave the ratio its Greek letter, φ, after the sculptor Phidias, while saying he doubted Phidias had ever used it.

The famous examples mostly fall apart when someone takes a ruler to them. The Parthenon’s plans are laid out in simple ratios such as four to nine. When Clement Falbo measured nautilus shells in a museum collection, their spirals widened about three times a turn⁵; a golden spiral widens almost seven. Leonardo drew his famous man in a circle and a square before he ever met Pacioli. And the Apple logo was drawn, its designer says, from a bowl of real apples.

The true places the golden ratio turns up are less famous, and far more surprising.

Look at the middle of a sunflower. Each new seed grows a fixed turn around from the last, about 137.5 degrees, an angle made from the golden ratio. Because φ is the number that fractions find hardest to pin down, seeds placed at that angle never line up behind one another, and the head packs tight with no gaps. Turn by half a degree more or less and the seeds fall into spokes. Physicists have even made the same pattern with magnetised drops of liquid in a dish, so it seems to be the simple result of things growing one after another, each pushing the last aside.

Nature is not tidy about it, though. In 2016 hundreds of volunteers grew sunflowers for a study named after Alan Turing, who had wondered about this before he died. They counted the spirals on 657 heads, and nearly one in five⁶ broke the famous pattern.

The numbers that go with it, 1, 2, 3, 5, 8, 13, each the sum of the two before, are named after Fibonacci, who wrote about them in 1202. But poets in India had found them centuries earlier, by counting the ways long and short beats can fill a line of verse.

In 1982 the chemist Dan Shechtman saw a crystal whose pattern, everyone agreed, could not exist. Its atoms were spaced in golden proportions. He was laughed at for years, and in 2011 he won the Nobel Prize⁷. And on any drive, a mile is very nearly 1.618 kilometres. That is pure coincidence, but it means 5 miles is about 8 kilometres, and 8 miles about 13.

We do not use the golden ratio because it is magic. We use it the way a composer chooses a key: once chosen, every part of the piece belongs to it.

It has two plain properties that make it good at this job. Cut a square from a golden rectangle and what is left is a smaller golden rectangle, and no other rectangle does that, so a page can be divided again and again and every piece keeps the family shape. And on a golden scale, each step is the two steps below it added together. Our type sizes work that way: 16 and 26 make 42, and 26 and 42 make 68.

That matters most on screens, because a screen never holds still. The same website is read on a phone in someone’s hand and on a television across the room. Most websites are drawn twice, once as wide as a laptop and once as narrow as a phone, and everything in between is left to stretch until it snaps. We would rather keep one proportion at every width, so each division of the screen takes the same share of the space it divides, whether the screen is a watch or a wall. A computer checks the rule every time we change the site, so nobody has to remember it and nobody can quietly nudge it out of line.

Book designers have done something like this for centuries, though not usually with this ratio. Many old books are two to three, and the great typographer Jan Tschichold, who studied their pages closely, called the familiar five to eight “no more than an approximation” of the golden section. We like that honesty, and we have tried to keep it.

We chose one proportion and we keep it. That turns a thousand small guesses into a single decision, made once.

It is a kindness to the people who read the page and to the people who build it. The margin, the picture and the type all come from the same family, and anyone who works on the site can see why each size is what it is.

The things that matter most to reading do not come from the golden ratio at all: letters large enough for your eyes, lines neither too long nor too short, and space between the lines that suits the size of the letters. Those come from measuring, and we measure them.

We still mean to ask people directly, by showing them two versions of the same page and asking which holds together better, and to publish what they tell us, whatever it is.

The rule is not there to replace the eye. It is more like a well-made frame: it gives the work a steady edge, so the eye is free to do the rest.

We study art and aesthetics because artists are constantly trying to translate the world around them into something for humanity. This is my attempt at reflecting nature in my own work.

Robert Duebelbeis, Good Citizens