What do these have in common:

• the shapes of spiral galaxies;
• the branching of leaves on a stem;
• the flight path of a diving falcon;
• the breeding of rabbits;
• the spiral shapes of nautilus shells and other mollusks;
• the way black holes change from one “phase” to another; and
  
• Salvador Dali’s painting Sacrament of the Last Supper

What shared thread connects the petal arrangement in a red rose with the architecture of Le Corbusier? The answer is, all these phenomena share a close association with a single, extraordinary number. No wonder the number in question has earned the name “golden ratio” and “divine proportion.”

The golden ratio - also known as “golden section,” and “golden number,” - is hardly, by itself, a novel concept. The systematizer of Greek geometry, Euclid, who taught in Alexandria around 300 B.C., defined the number in Elements, his famous work on geometry and number theory. But Euclid's definition was entirely geometric and betrayed not the slightest acquaintance with the role of the golden ratio in the natural world. In fact, it was nothing more than a modestly amusing way for geometers to divide a line into two unequal parts. Little did Euclid know that his innocent-looking division would preoccupy mathematicians, physicists, botanists, psychologists, and artists for the next few millennia.

Euclid's number (the name “golden ratio” was applied many centuries later) emerges from geometry in the following way: Take any line segment and divide it into two parts, in such a way that the longer part of the line segment (AC) is in the same proportion to the shorter part (CB) as the entire line segment (AB) is to the longer part (AC). The ratio in question (AC/CB) is the golden ratio [see diagram below].

Some simple high school algebra shows that the golden ratio is equal to the never-ending, never-repeating number 1.6180339887..., commonly denoted by the Greek letter phi, or Φ.

Like pi, the number phi is an irrational number, one that cannot be expressed as a ratio of two whole numbers, such as 3/1, 3/2, 5/7, or 23/39. In fact, phi is mathematically the “most irrational” number, in the sense that, if you try to approximate it as what is known as a continued fraction (one in which fractions are added in the denominator ad infinitum), you find that the approximation converges on it more slowly than continued-fraction approximations do for any other irrational number.

The number phi would have remained in the relative obscurity of pure mathematics were it not for its propensity to pop up where least expected. Take, for instance, the head of a sunflower. The florets form various clockwise and counterclockwise spiral patterns, intertwined and crisscrossing but otherwise unmistakable to the eye. Each floret arises in the center of the sunflower and gets pushed outward by its successors; the spiral patterning is an outcome of the way the florets are most easily and efficiently packed as they grow. The number of clockwise spirals and the number of counterclockwise spirals vary, depending on the size of the sunflower. Usually you find 55 twisting one way and 34 the other, but you may find 89 and 55, or 144 and 89. Even 233 and 144 has been reported. Amazingly, if you calculate these as ratios (55/34, 89/55, 144/89, 233/144), you find that they get closer and closer to the value of the golden ratio phi!

The patterning of sunflowers is closely related to one of the discoveries made in 1837 by two French brothers, Auguste and Louis Bravais. Auguste, a crystallographer, and Louis, a botanist, observed that as new leaves are put forth from the tip of many growing plants, each new leaf advances by an angle of roughly 137.5 degrees from the preceding leaf, around the circumference of the stem. That angle is what you get if you divide the number of degrees in a complete circle, 360, by the number phi, and then subtract the result from 360.

But why should the leaves of a plant arrange themselves in a pattern that is based on a number derived from the division of a line? If the angle between the leaves is, say, 90 degrees (which is equal to 360/4), or any other simple fraction of 360 degrees, the leaves will align one above the other on the stem, leaving large spaces unfilled. (In the case of 90 degrees, they will make four lines along the stem.) Such an arrangement would probably be undesirable for the plant, because overlapping leaves would shield one another from the light they need. By arranging themselves according to an angle determined by phi, the leaves can fill the spaces in the most efficient way possible, with the least amount of overlap.

Botany is hardly the only context in which the golden ratio appears. Take the so-called golden rectangle, in which the ratio of the length to the width is equal to phi. If you snip off a square from the rectangle, the rectangle that remains is also a golden rectangle. You can continue this process of snipping off squares ad infinitum, generating smaller and smaller golden rectangles. No other rectangle gives rise to the same shape as you snip off successive squares. If you then connect the successive points where the whirling squares cut the sides of the rectangles, you get a curve known as a logarithmic spiral [see illustration below].

The name follows from an observation by the seventeenth-century Swiss mathematician Jakob Bernoulli, of the way in which the radius grows when one moves around the curve.

Bernoulli recognized that the logarithmic spiral does not alter its shape as its size increases, a property known as self-similarity. For that reason, Bernoulli noted, the spiral “may be used as a symbol, either of fortitude and constancy in adversity, or of the human body, which after all its changes, even after death, will be restored to its exact and perfect self.” He asked to have the spiral engraved on his tombstone - but, sadly, ignorance prevailed, and the tombstone artist carved only the simple coil (the shape formed by, say, a roll of paper towels) known as the Archimedean spiral.

Another intriguing property of the logarithmic spiral is that it is equiangular: if you draw a straight line from the center, or pole, to any point on the spiral curve, the line always cuts the curve at precisely the same angle. Falcons bank on this property when attacking their prey. Vance A. Tucker, a biologist at Duke University in Durham, North Carolina, studied falcons for many years and discovered that they usually follow a slightly curved trajectory to their victims, rather than plummeting in a straight line. Tucker eventually realized that the falcons’ trajectory could be a consequence of keeping the fovea of one or the other eye, the most acute part of their vision, locked onto their target. To make use of the fovea during a straight downward plunge, the falcons would have to cock their heads some forty degrees to one side or the other. But Tucker showed in wind-tunnel experiments that cocking the head would slow the falcons down considerably. By keeping their heads straight while keeping their target in view from the most advantageous angle, the falcons naturally follow the curve of a (highly drawn-out) logarithmic spiral.

Nature just loves logarithmic spirals. You can find them in phenomena ranging from the shell of the chambered nautilus to hurricanes and spiral galaxies. Sometimes, as in the case of the nautilus, they are a natural outcome of a pattern of additive growth. And it is through that pattern that the golden ratio is intimately related to the Fibonacci sequence, a celebrated series of numbers discovered by the early thirteenth-century Italian mathematician Leonardo of Pisa, known as Fibonacci.

In his book Liber abaci (Book of the Abacus), published in 1202, Fibonacci posed the following fanciful problem about the breeding of rabbits: “A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair, which from the second month on becomes productive?”

The solution to the problem is fairly simple. Start with one pair of baby rabbits.

After a month you still have only the one pair of rabbits, now nearing maturity. In the third month, however, you have two pairs of rabbits (the original pair, plus their first two babies). Come back in another month and you have three pairs, because the first pair has generated another set of babies. In the fifth month you have five pairs (because the first pair of babies has become old enough to reproduce). And so on.

You end up with the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, and so on, in which each term (from the third on) is equal to the sum of the two preceding terms. The sequence was named the Fibonacci sequence by the nineteenth-century French mathematician Edouard Lucas. For the sake of historical accuracy, one should note that this sequence of numbers actually appeared even earlier than Fibonacci, in a rule for the construction of meter in a category of Sanskrit poems known as matravrittas. Indian poets wrote about the rule in detail before Fibonacci was born, but Western mathematicians were unaware of their contributions until the appearance of a 1985 article by Parmanand Singh, a mathematician then at Raj Narain College in Hajipur, India.

You may have noticed that some of the numbers in the Fibonacci series have already been mentioned: they are the same as the numbers of clockwise and counterclockwise spirals appearing in sunflowers. And recall that the ratios of the numbers of spirals were good approximations of phi. It turns out that if you calculate the ratios of successive Fibonacci numbers [approximated to the sixth decimal place in the table at right], the ratios oscillate about phi but also converge on it as you go farther out along the sequence.

Thus Fibonacci numbers are a kind of golden ratio in disguise, and they, too, pop up in the most unexpected places. One is in the microtubules of an animal cell, which are hollow cylindrical tubes of a protein polymer. Together they make up the cytoskeleton, a structure that gives shape to the cell and also appears to act as a kind of cell “nervous system.” Each mammalian microtubule is typically made up of thirteen columns, arranged in five right-handed and eight left-handed structures (5, 8, and 13 are all Fibonacci numbers). Furthermore, occasionally one finds double microtubules with an outer envelope made up—you guessed it—of 21 columns, the next Fibonacci number. The precise reason that the Fibonacci numbers show up in microtubules is not clear, but some investigators have argued that microtubules structured this way are more efficient than other possible structures are as “information processors.” Because these sets of numbers are so small, however, the apparent connection with the Fibonacci series may be coincidental.

Turning from the microscopic to nature on a large scale, one finds that the spiral arms of many disk-shaped galaxies are often close to logarithmic spirals [see photograph below]. The spiral arms stand out because that is where many stars are being born, and younger stars, on average, are brighter than older ones. But how do such spiral arms retain their shape over long periods of time? The reason this question is an astrophysical puzzler is that a galaxy does not rotate about its center like a disk of solid material, in which all parts simultaneously make a complete circuit. Instead, the closer to the center the stars or other matter lie, the faster they rotate. A spiral arm made up of some fixed group of bright stars should quickly get “wound up” - but that would imply that spiral galaxies were much rarer than they are observed to be.

The explanation is that the spirals are not structures of connected material streaming out from the center of a galaxy, as they might appear. Instead, they are the result of waves of gas compression sweeping through the disk. Where gas is compressed, the birth of new stars is triggered. Because matter is not uniformly distributed throughout the galaxy, the waves sustain a spiral effect as a kind of interference pattern. The golden ratio makes an unexpected appearance even in the thermodynamics of certain black holes. Black holes can be either non-rotating (have no angular momentum), or spinning. Spinning black holes (called Kerr black holes, after the New Zealander physicist Roy Kerr) can exist in two states, one in which they heat up when they lose energy (negative specific heat) and one in which they cool down. They also can undergo a phase transition (similar to the freezing of water) from one state to the other. The transition can take place only when the black hole reaches a state in which the square of its mass is precisely equal to phi times the square of its angular momentum (in the appropriate units).

This seemingly magical appearance of phi stems from another unique mathematical property of the golden ratio: its square can be obtained simply by adding 1 to phi (you can check that statement with a pocket calculator).

Many books claim that if you draw a rectangle around the face of Leonardo da Vinci’s Mona Lisa, the ratio of the height to width of that rectangle is equal to the golden ratio. No documentation exists to indicate that Leonardo consciously used the golden ratio in the Mona Lisa’s composition, nor to where precisely the rectangle should be drawn. Nevertheless, one has to acknowledge the fact that Leonardo was a close personal friend of Luca Pacioli, who published a three-volume treatise on the golden ratio in 1509 (entitled Divina Proportione).

Another painter, about whom there is very little doubt that he actually did deliberately include the golden ratio in his art, is the surrealist Salvador Dali. The ratio of the dimensions of Dali's painting Sacrament of the Last Supper is equal to the Golden Ratio. Dali also incorporated in the painting a huge dodecahedron (a twelve-faced Platonic solid in which each side is a pentagon) engulfing the supper table. The dodecahedron, which according to Plato is the solid “which the god used for embroidering the constellations on the whole heaven,” is intimately related to the golden ratio - both the surface area and the volume of a dodecahedron of unit edge length are simple functions of the golden ratio.

These two examples are only the tip of the iceberg in terms of the appearances of the golden ratio in the arts. The famous Swiss-French architect and painter Le Corbusier, for example, designed an entire proportional system called the “Modulor,” that was based on the golden ratio. The Modulor was supposed to provide a standardized system that would automatically confer harmonious proportions to everything, from door handles to high-rise buildings.

In these and countless other ways, the golden ratio triggers the feeling of amazement that Einstein regarded as essential for all intellectual endeavors. In Einstein’s words:

“The fairest thing we can experience is the mysterious. It is the fundamental emotion which stands at the cradle of true art and science. He who knows it not and can no longer wonder, no longer feel amazement, is as good as dead, a snuffed-out candle.”

Some Additional Bibliography

• Dantzig, T.  1954, Number: The Language of Science (New York: The Free Press)
• Dunlap, R. A.  1997, The Golden Ratio and Fibonacci Numbers (Singapore: World Scientific)
• Fischler, R.  1981, “On the Application of the Golden Ratio in the Visual Arts,” Leonardo, 14, 31
• Gies, J. & Gies, F.  1969, Leonardo of Pisa and the New Mathematics of the Middle Ages (New York: Thomas Y. Crowell Company)
• Howat, R.  1983, Debussy in Proportion (Cambridge: Cambridge University Press)
• Livio, M.  2003, Hitukh Hazahav (Tel-Aviv: Aryeh Nir)
• MacKinnon, N.  1993, “The Portrait of Fra Luca Pacioli,” Mathematical Gazette, 77, 130
• McManus, I. C.  1980, “The Aesthetics of Simple Figures,” British Journal of Psychology, 71, 505