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
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
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 common 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-gravitating 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
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
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..."