杏吧原创

IT鈥橲 inexplicable: people seem to like some rectangles more than others. Present them with a range of choices, from fat and square to long and thin, and they will pick out as most 鈥減leasing鈥� the rectangle with side lengths closest to a particular ratio.

This number 鈥� known as the golden ratio, or phi 鈥� has been celebrated as a fundamental of aesthetics and is believed to have been used all over the place, from Greek architecture to the framing of the Mona Lisa鈥檚 face. But phi crops up in the sciences too. And its latest appearance, in the properties of some particularly weird metals, has recently been published in Physical Review B, a journal of solid-state physics not noted for its contribution to art history.

Indeed, the art world has long been losing its traditional grip on phi. We now know that the golden ratio crops up in the arrangement of leaves on plant stems, the shape of a sunflower鈥檚 seed head and a seashell鈥檚 spiral, and even the properties of spinning black holes. It鈥檚 simply everywhere in our Universe. What we don鈥檛 know is why.

The golden ratio was first described explicitly by the Greek mathematician Euclid of Alexandria around 300 BC, although it was probably known to the followers of Pythagoras two centuries earlier. Euclid defined it in terms of a line divided into two unequal segments (see Graphic). The ratio of the longer segment to the smaller one is said to be in the golden proportion if it is equal to the ratio of the whole line to the longer segment. Numerically, the golden ratio is equal to (1 + 鈭�5)/2, or 1.6180339887鈥�.

But this mundane definition belies the ratio鈥檚 unique and surprising attributes. 鈥淚t has an almost endless list of peculiar properties,鈥� says Mario Livio, head of science at the Space Telescope Science Institute in Baltimore. For instance, you can square it simply by adding 1 and find its reciprocal simply by subtracting 1. As a result of this property, if you take a golden rectangle 鈥� one whose length and width are in the golden ratio 鈥� and snip from it a square, you are left with another golden rectangle.

Here鈥檚 another enigma. Choose any two numbers and form a third number by adding the first and second. Then form a fourth by adding the the third and the second, a fifth by adding the third and the fourth, and so on. For example, if you started with 7 and 11, you鈥檒l get 7, 11, 18, 29, 47, 76鈥�. Once you have about 20 numbers, divide the 20th by the 19th, and you鈥檒l find the answer is almost exactly equal to the golden ratio.

You may recognise the connection here with the mathematical Fibonacci sequence of numbers. It begins 1, 1, 2, 3鈥� and generates each new term by adding the preceding two. The sequence describes the proportions of a seashell鈥檚 spiral and the arrangement of seeds on a sunflower head.

Phi owes its favour with artists and architects to an Italian Franciscan friar and mathematician Luca Pacioli. In the 15th century Pacioli published a three-volume treatise called The Divine Proportion, a tract on religion, philosophy and aesthetics. He used the fact that the digits in the golden ratio鈥檚 decimal expansion never repeat as an analogy for the incomprehensibility of God. 鈥淭he book is crucial,鈥� says Livio. 鈥淚t brought the golden ratio to the attention of the wider world of artists, musicians and architects.鈥�

Since Pacioli鈥檚 time, numerous painters, builders and musicians have used the golden ratio explicitly in their work. Some of the most prominent examples include the composers Debussy and possibly Bart贸k, and the architect Le Corbusier. But phi鈥檚 hold on the artistic world may not be as strong as once thought.

For a start, the argument that phi is the most aesthetic proportion seems to be something of a myth. 鈥淭here is no conclusive scientific proof that the human mind reacts in any special way to shapes in the golden proportion,鈥� says Livio. He has even dared to throw cold water on claims that Leonardo da Vinci painted the face of the Mona Lisa to fit inside a golden rectangle. Although da Vinci was a friend of Pacioli鈥檚 and would certainly have been aware of the golden ratio, whether he actually used it is open to question. 鈥淚t all depends on where you draw the rectangle, and that鈥檚 not at all clear,鈥� Livio says.

George Markowsky, a mathematician at the University of Maine in Orono, reckons the claims that the Egyptians used the golden ratio in the construction of the Great Pyramid of Cheops, and that the Greeks used it in the construction of the Parthenon, are more fantasy than fact. 鈥淚 heard the claims about the golden ratio being used in the Parthenon and so on, but nobody would say exactly where,鈥� he says. When he studied the measurements, Markowsky found no support for any of the claims. 鈥淢uch of what is said about the golden ratio in art, architecture and literature is seriously misleading,鈥� he concludes.

However, there鈥檚 no doubt that the golden ratio does appear in nature and science 鈥� and in the most surprising places. Take phyllotaxis, for example, the arrangement of leaves on a vertical plant stem. As each new leaf grows, it radiates out from the stem offset at an angle to the one immediately below. Remarkably, the most common offset angle turns out to be 137.5掳, which is exactly what you get when you divide 360掳 into two angles in the golden ratio: 137.5掳 and 222.5掳.

But why should this 鈥済olden angle鈥� crop up in phyllotaxis? It鈥檚 all to do with efficiency. Each new leaf at the top of the stem must collect sunlight without throwing the leaves below it into too much shadow. So ideally, a plant must arrange its leaves in such a way that as many as possible can spiral around the stem before a new leaf sprouts immediately above a lower one and obscures its light.

If the leaves were offset by 120掳, say, then looking at the plant from above you鈥檇 see all the leaves line up in three neat columns with big gaps between these columns. If the angle between them were 50掳 there鈥檇 be more than three columns, but there would still be gaps and, after a fairly small number of leaves up the stem, there鈥檇 be a leaf directly over one below it. But with an offset of 137.5掳, the plant minimises the gaps and maximises the number of leaves that it can sprout without compromising its light-gathering efficiency.

The golden ratio also crops up in some of the toughest areas of science. Consider the growth of 鈥渜uasicrystals鈥�, crystals that are not truly regular but which have fivefold symmetry so that they look the same when rotated by a fifth of a full turn. Since their discovery in 1984, many researchers have begun to grow them and examine their strange properties.

Tanhong Cai at the Brookhaven National Laboratory in New York state has studied high-magnification images of the surfaces of two such crystals 鈥� an aluminium-copper-iron alloy and an aluminium-palladium-manganese alloy. They reveal flat terraces punctuated by abrupt vertical steps. The steps come in two predominant sizes. And the ratio of the two step heights? You鈥檝e guessed it. This latest manifestation of the golden ratio was found only this year.

But phi isn鈥檛 restricted to Earth-based phenomena. The golden ratio is even out there in black holes. In 1989, Paul Davies of the University of Adelaide discovered that the thermodynamics of rotating black holes is inextricably linked with phi.

Most things have a positive 鈥渟pecific heat鈥�: they get colder as they release energy. But a rotating black hole can have a negative specific heat, so it gets hotter as it releases energy. The properties that determine whether the black hole鈥檚 specific heat is positive or negative are its mass and its speed of rotation, characterised by a 鈥渟pin parameter鈥�. Davies discovered that a rotating black hole flips from a negative to a positive specific heat when the square of its mass divided by the square of its spin parameter is equal to the golden ratio.

Why should the golden ratio be associated with black holes? 鈥淚t鈥檚 a complete enigma,鈥� says Livio. 鈥淭here may be a deep reason or it may be just coincidence.鈥�

Markowsky argues for the latter answer, and is not impressed by the notion that there is some deep 鈥渃osmic plan鈥� behind the ubiquity of the golden ratio. It is an interesting number, he admits, but so are many other numbers. 鈥淚 am all for people showing interesting uses that it has, just as I like &pgr;, e, 0, 1 and so on. But I don鈥檛 have much sympathy with making a cult of the number and somehow claiming that it is more significant than any other.鈥�

Nevertheless, Livio thinks the golden ratio鈥檚 scientific odyssey is only just beginning. Even if its artistic heritage crumbles in their hands, scientists will find the golden ratio in many more places 鈥� it appears to rule the Universe.