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June 2004

Good stories, pity they're not true

The enormous success of Dan Brown's novel The Da Vinci Code has introduced the famous Golden Ratio (henceforth GR) to a whole new audience. Regular readers of this column will surely be familiar with the story. The ancient Greeks believed that there is a rectangle that the human eye finds the most pleasing, and that its aspect ratio is the positive root of the quadratic equation:

x2 - x - 1 = 0

You are faced with this equation when you try to determine how to divide a line segment into two pieces such that the ratio of the whole line to the longer part is equal to the ratio of the longer part to the shorter. The answer is an irrational number whose decimal expansion begins 1.618.

Having found this number, the story continues, the Greeks then made extensive use of the magic number in their architecture, including the famous Parthenon building in Athens. Inspired by the Greeks, future generations of architects likewise based their designs of buildings on this wonderful ratio. Painters did not lag far behind. The great Leonardo Da Vinci is said to have used the Golden Ratio to proportion the human figures in his paintings - which is how the Golden Ratio finds its way into Dan Brown's potboiler.

It's a great story that tends to get better every time it's told. Unfortunately, apart from the fact that Euclid did solve the line division problem in his book Elements, there's not a shred of evidence to support any of these claims, and good reason to believe they are completely false, as University of Maine mathematician George Markowsky pointed out in his article "Misconceptions About the Golden Ratio", published in the College Mathematics Journal in January 1992. But with such a wonderful story, which marries some decidedly accessible pure mathematics with aethestics, architecture, and painting - a high school math teacher's dream if ever there were one - the facts have had little impact.

But being aware that few people will take note of what I say has never stopped me before. (I was, after all, a department chair in a college mathematics department for four years and a college dean for another eight.) So let's try to separate the fact from the fiction.

First, what do we know for sure about the Golden Ratio? As mentioned above, Euclid showed how to calculate it, but his interest seemed more that of mathematics than visual aesthestics or architecture, for he gave it the decidedly unromantic name "extreme and mean ratio". The term "Divine Proportion," which is oten used to refer to GR, first appeared with the publication of the three volume work by that name by the 15th century mathematician Luca Pacioli. Calling GR "golden" is even more recent: 1835, in fact, in a book written by the mathematician Martin Ohm (whose physicist brother discovered Ohm's law).

It is also true that the Golden Ratio is linked to the pentagram (five-pointed star), to the five Platonic solids, to fractal geometry, to certain crystal structures, and to Penrose tilings. So far so good.

The oft repeated claim (actually, all claims about GR are oft repeated) that the ratios of successive terms of the Fibonacci sequence tend to GR is also correct. The Fibonacci sequence, you may recall, is generated by starting with 0, 1 and repeatedly applying the rule that each new number is equal to the sum of the two previous numbers. So 0+1 = 1, 1+1 = 2, 1+2 = 3, 2+3 = 5, etc., giving the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ... The sequence of successive ratios of the numbers in this sequence, namely 1/1 = 1; 2/1 = 2; 3/2 = 1.5; 5/3 = 1.666... ; 8/5 = 1.6; 13/8 = 1.625; 21/13 = 1.615...; 34/21 = 1.619...; 55/34 = 1.6176...; 89/55 = 1.6181; ..., does indeed tend to GR. As I'll explain momentarily, this is a key part of the explanation of why the Fibonacci numbers keep appearing in flowers and plants - which they do.

For instance, if you count the number of petals in most flowers you will find that the answer is a Fibonacci number. For example, an iris has 3 petals, a primrose 5, a delphinium 8, ragwort 13, an aster 21, daisies 13, 21, or 34, and Michaelmas daisies 55 or 89 petals. All Fibonacci numbers.

Again, if you look at a sunflower, you will see a beautiful pattern of two spirals, one running clockwise, the other counterclockwise. Count those spirals, and for most sunflowers you will find that there are 21 or 34 running clockwise and 34 or 55 counterclockwise, respectively - all Fibonacci numbers. Less common are sunflowers with 55 and 89, with 89 and 144, and even 144 and 233 in one confirmed case. Other flowers exhibit the same phenomenon; the wildflower Black-Eyed Susan is a good example. Similarly, pine cones have 5 clockwise spirals and 8 counterclockwise spirals, and the pineapple has 8 clockwise spirals and 13 going counterclockwise.

Finally, if you take a close look at the way leaves are located on the stems of trees and plants, you will see that they are located on a spiral that winds around the stem. Starting at one leaf, count how many complete turns of the spiral it takes before you find a second leaf directly above the first. Let P be that number. Also count the number of leaves you encounter (excluding the first one itself). That gives you another number Q. The quotient P/Q is called the divergence of the plant. (The divergence is characteristic for any particular species.) If you calculate the divergence for different species of plants, you find that both the numerator and the denominator are usually Fibonacci numbers. In particular, 1/2, 1/3, 2/5, 3/8, 5/13, and 8/21 are all common divergence ratios. For instance, common grasses have a divergence of 1/2, sedges have 1/3, many fruit trees (including the apple) have a divergence of 2/5, plantains have 3/8, and leeks come in at 5/13.

Although many of these observations were made a hundred year or more ago, it was only recently that mathematicians and scientists were finally able to figure out what is going on. It's a question of Nature being efficient.

For instance, in the case of leaves, each new leaf is added so that it least obscures the leaves already below and is least obscured by any future leaves above it. Hence the leaves spiral around the stem. For seeds in the seedhead of a flower, Nature wants to pack in as many seeds as possible, and the way to do this is to add new seeds in a spiral fashion.

As early as the 18th century, mathematicians suspected that a single angle of rotation can make all of this happen in the most efficient way: the Golden Ratio (measured in number of turns per leaf, etc.). However, it took a long time to put together all the pieces of the puzzle, with the final step coming in the early 1990s.

The worst kind of angle for efficient growth would be a rational number of turns, eg. 2 turns, or 1/2 a turn, or 3/8 of a turn, since they will soon lead to a complete cycle. Mathematically, a turn through an irrational part of a circle will never cycle, but in practical terms it could eventually come close. What angle will come least close to a cycle? Maximum efficiency will be achieved when the angle is "furthest away" from being a rational. But what exactly does that mean? The appropriate way (via s vis plant growth) to measure how far removed from being rational an irrational number is, is to look at its continued fraction expansion. For GR, this is:

Rational numbers have a finite continued fraction. That unending, constant sequence of 1's in the continued fraction for GR says that, measured in terms of continued fractions, GR is the irrational number furthest removed from being rational. And that's the mathematical reason why Nature favors GR as her growth ratio. The Fibonacci numbers appear because the number of leaves, spirals, etc. are whole numbers, and (because of the ratio limit property mentioned above) the Fibonacci numbers are the best whole number approximations to a GR growth.

For other examples of the appearance of the Golden Ratio in Nature, the growth of the Nautilus shell is governed by the Golden Ratio, as is the path followed by a Pergrine falcon when it swoops down to catch its pray. In these cases, the explanation is that the GR is closely related to the logarithmic spiral, the spiral that turns by a constant angle along its entire length, making it everywhere self-similar.

As the Nautilus grows, it has repeated need to enlarge its living quarters. Since the creature does not change shape, rather simply grows larger, the most efficient way to do this is for its shell to grow in the self-similar form of a logarithmic spiral.

The falcon must keep the prey in its sight all the time, but, although its eyes are razor sharp, they are fixed in its head, one on either side. So what the creature does is swivel its head to one side, by an angle of about 40o, and fix the prey in one eye. Keeping its head fixed at that 40o angle, the falcon then dives in a way that keep the prey in view in that one eye. The fixed angle of the head results in the bird following an equi-angular spiral path that converges on the prey.

So much for the good (i.e., true) stuff. Now for those many, many myths about GR that continue to do the rounds. The issue here is not whether you can find GR somewhere. If you look hard enough you will be able to find any (reasonably sized) number almost anywhere. The question is whether there is more to it than mere numerology. Is there a good scientific explanation to show why GR appears (as with the examples from Nature mentioned above), or is there definite evidence that, say, a particular artist made deliberate use of GR in his or her work? If not, all you have is an unsubstantiated belief. You may as well believe in fairies.

First of all, whether or not the ancient Greeks felt that the Golden Ratio was the most perfect proportion for a rectangle, many modern humans do not. Numerous tests have failed to show up any one rectangle that most observers prefer, and preferences are easily influenced by other factors. As to the Parthenon, all it takes is more than a cursory glance at all the photos on the Web that purport to show the Golden Ratio in the structure, to see that they do nothing of the kind. (Look carefully at where and how the superimposed rectangle - usually red or yellow - is drawn and ask yourself: why put it exactly there and why make the lines so thick?)

Another claim is that if you measure the distance from the tip of your head to the floor and divide that by the distance from your belly button to the floor, you get GR. But this nonsense. First of all, you won't get exactly the number GR. You never can; GR is irrational, remember. But in the case of measuring the human body, there is a lot of variation. True, the answers will always be fairly close to 1.6. But there's nothing special about 1.6. Why not say the answer is 1.603? Besides, there's no reason to divide the human body by the navel. If you spend a half an hour or so taking measurements of various parts of the body and tabulating the results, you will find any number of pairs of figures whose ratio is close to 1.6, or 1.5, or whatever you want.

Then there is the claim that Leonardo Da Vinci believed the Golden Ratio is the ratio of the height to the width of a "perfect" human face and that he used GR in his Vitruvian Man painting. While there is no concrete evidence against this belief, there is no evidence for it either, so once again the only reason to believe it is that you want to. The same is also true for the common claims that Boticelli used GR to proportion Venus in his famous painting The Birth of Venus and that Georges Seurat based his painting The Parade of a Circus on GR.

Painters who definitely did make use of GR include Paul Serusier, Juan Gris, and Giro Severini, all in the early 19th century, and Salvador Dali in the 20th, but all four seem to have been experimenting with GR for its own sake rather than for some intrinsic aesthetic reason. Also, the Cubists did organize an exhibition called "Section d'Or" in Paris in 1912, but the name was just that; none of the art shown involved the Golden Ratio.

Then there are the claims that the Egyptian Pyramids and some Egyptian tombs were constructed using the Golden Ratio. There is no evidence to support these claims. Likewise there is no evidence to support the claim that some stone tablets show the Babylonians knew about the Golden Ratio, and in fact there is good reason to conclude that it's false.

Turning to more modern architecture, while it is true that the famous French architect Corbusier advocated and used the Golden Ratio in architecture, the claim that many modern buildings are based on Golden Rectangles, among them the General Secretariat building at the United Nations headquarters in New York, seems to have no foundation. By way of an aside, a small (and not at all scientific) survey I carried out myself a few months ago revealed that all architects polled knew about the GR, and all believed that other architects used the GR in their work, but none of them had ever used it themselves. Make whatever inference you wish.

Music too is not without its GR fans. Among the many claims are: that some Gregorian chants are based on the Golden Ratio, that Mozart used the Golden Ratio in some of his music, and that Bartok used GR in some of his music. All those claims are without any concrete support. Less clear cut is whether Debussy used the Golden Ratio in some of his music. Here the experts don't agree on whether some GR patterns that can be discerned are intended or spurious.

Poetry too is not immune, but here there is a refreshing surprise in store for us. Whereas the claim that the Roman poet Vergil based the meter of his poem Aeneid on the Golden Ratio has no support, it really is true that some 12th Century Sanskrit poems have a meter based on the Fibonacci sequence (and hence related to the Golden Ratio).

I could go on, as there are many more examples, ranging from the sacred (eg. the dimensions of the Ark of the Covenant) to the profane (such as, predicting the behavior of the stock market), all of which, on close examination, turn out to be without any supporting evidence whatsoever. Despite the lack of evidence, however, and in some cases in the face of evidence to the contrary, each claim seems to attract its own band of devotees, who will not for a moment entertain the possibility that their cherished beliefs are not true. Consequently, not only is GR a very special number mathematically - all of its genuine appearances in mathematics and Nature show that - it also has enormous cultural significance as the number that most people have the greatest number of false beliefs about. Now there's a GR fact that has plenty of supporting evidence.

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For more details on the Golden Ratio, including evidence to support many of the claims I have made above, see the Markowsky article mentioned earlier, as well as the excellent book The Golden Ratio: The Story of PHI, the World's Most Astonishing Number, by Mario Livio. Also worth a visit is Ron Knott's excellent website Fibonacci Numbers and the Golden Section, at the University of Surrey in England.

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Devlin's Angle is updated at the beginning of each month.

Mathematician Keith Devlin ( [email protected]) is the Executive Director of the Center for the Study of Language and Information at Stanford University and The Math Guy on NPR's Weekend Edition. Devlin's most recent book is Sets, Functions, and Logic: an Introduction to Abstract Mathematics (Third Edition), published by Chapman and Hall in 2003.

http://www.maa.org/devlin/devlin_06_04.html