It links art, music and even architecture.
Marcus Chown on an enigmatic number
(from The Guardian, January 16, 2003)
Think of any two numbers. Make a third by adding the first and second, a fourth by adding the second and third, and so on. When you have written down about 20 numbers, calculate the ratio of the last to the second from last. The answer should be close to 1.6180339887…
What’s the significance of this number? It’s the “golden ratio” and, arguably, it crops up in more places in art, music and so on than any number except pi. Claude Debussy used it explicitly in his music and Le Corbusier in his architecture. There are claims the number was used by Leonardo da Vinci in the painting of the Mona Lisa, by the Greeks in building the Parthenon and by ancient Egyptians in the construction of the Great Pyramid of Khufu.
What makes the golden ratio special is the number of mathematical properties it possesses. The golden ratio is the only number whose square can be produced simply by adding 1 and whose reciprocal by subtracting 1. If you take a golden rectangle – one whose length-to-breadth is in the golden ratio – and snip out a square, what remains is another, smaller golden rectangle. The golden ratio is also difficult to pin down: it’s the most difficult to express as any kind of fraction and its digits – 10 million of which were computed in 1996 – never repeat.
It was this elusive nature that led the 15th-century Italian friar and mathematician Luca Pacioli to equate the golden ratio with the incomprehensibility of God. Although Euclid defined it around 300 BC, and the followers of Pythagoras probably knew of it two centuries earlier, it was Pacioli’s three-volume treatise, The Divine Proportion, that was crucial in disseminating the golden ratio beyond the world of mathematics.
Da Vinci was a friend of Pacioli’s and almost certainly would have read the book, hence the claim that he painted the face of the Mona Lisa to fit inside a hypothetical golden rectangle.
“Of course, it all depends on how you draw the rectangle!” says Mario Livio, who has written a book called The Golden Ratio and who is head of science at Baltimore’s Space Telescope Science Institute.
The appeal of the divine proportion to the human eye and brain has been scientifically tested. Dozens of psychological tests, beginning with those of Gustav Fechner in the 19th century, have shown that, when subjects are presented with a range of rectangles, they invariably pick out as most pleasing ones whose sides are in the golden ratio.
But the most surprising thing is that a number deemed aesthetically pleasing by human beings also crops up in nature and science. Take the arrangement of leaves on the stem of a plant. As each new leaf grows, it does so at an angle offset from that of the leaf below. The most com mon angle between successive leaves is 137.5 – the golden angle. Why? Because 137.5 = 360 – 360/G, where G is the golden ratio. Why does the golden ratio play a role in the arrangement of leaves? It’s all down to the “irrationality” of the number. Irrational numbers are ones that cannot be expressed as the ratio of two whole numbers – for instance, 5/2.
“The golden ratio is arguably the most irrational of all irrational numbers,” says Livio. This can be said more precisely. Irrational numbers can be expressed as continued fractions – basically an infinite series of ever-diminishing terms. As each successive term is added, the continued fraction converges towards a single value.
“The golden ratio is the slowest of all continued fractions to converge,” says Livio. This turns out to be the key property. A new leaf must collect sunlight without throwing the leaves below it into too much shadow. A plant must arrange its leaves in such a way that the greatest number can spiral around the stem before a new leaf sprouts immediately above a lower one – that is offset at 360.
“What better way to do this than to choose an angle between leaves based on a number that takes the longest to converge?” says Livio.
The golden ratio also crops up in the hard sciences. Take the growth of “quasi-crystals”. These have “five-fold symmetry”, which means they make a pattern that looks the same when rotated by multiples of one-fifth of 360 . In the 1990s, physicists in Switzerland and the US imaged the microscopic terrain of the surface of such crystals. They found flat “terraces” punctuated by abrupt vertical steps. The steps come in two predominant sizes. The ratio of the two step heights? The golden ratio!
Even pythagoreans may have known of the association of the golden ratio with five-fold symmetry. The symbol of their cult was the five-pointed star, and the ratio of the length of the side of each triangular point to its projected base is the golden ratio.
Perhaps the most surprising place the golden ratio crops up is in the physics of black holes, a discovery made by Paul Davies of the University of Adelaide in 1989. Black holes and other self-gravitat ing bodies such as the sun have a “negative specific heat”. This means they get hotter as they lose heat. Basically, loss of heat robs the gas of a body such as the sun of internal pressure, enabling gravity to squeeze it into a smaller volume. The gas then heats up, for the same reason that the air in a bicycle pump gets hot when it is squeezed.
Things are not so simple, however, for a spinning black hole, since there is an outward “centrifugal force” acting to prevent any shrinkage of the hole. The force depends on how fast the hole is spinning. It turns out that at a critical value of the spin, a black hole flips from negative to positive specific heat – that is, from growing hotter as it loses heat to growing colder. What determines the critical value? The mass of the black hole and the golden ratio!
Why is the golden ratio associated with black holes? “It’s a complete enigma,” Livio confesses. Shakespeare said it all: “There are more things in heaven and earth…”
· Marcus Chown’s book, The Universe Next Door: 12 revolutionary ideas from the cutting edge of science (Headline), is out in paperback