Golden Ratio - how ubiquitous is it?

I recently read "The Da Vinci Code" by Dan Brown, and was reminded of the so called "power of Φ (Phi)". The story begins with the curator of Louvre Museum leaving a code scrawled on the museum floor just before he was murdered. The code has a jumbled set of numbers, which when unscrambled gives the first few elements "1-1-2-3-5-8-13-21" of the famous Fibonacci series, discovered by Leonardo Fibonacci in the 12th century. The obvious thing about this series is that each number is the sum of the preceding two numbers. Its less obvious feature is that, as the series progresses, the ratio of a number and its immediate predecessor approaches another famous mathematical entity, the Golden Ratio Φ = (√5+1)/2 = 1.61803398874989...(the 1.5 billionth significant digit can be found here). The golden ratio has a much longer history, and is believed to have been first defined by Euclid in 300 BC. Of course, mere deciphering of the numbers does not reveal the cause or the perpetrator of the murder in the story; there are wheels within wheels, or rather, codes within codes, and you can read all about them in the rest of the 450-page novel. My interest here is in the claim often made about the surprising ubiquity of Φ in nature around us.

In the story itself, the male protagonist Robert Langdon recalls his favorite Harvard lecture in which he talks to a bunch of thoroughly impressed students about the instances where this "divine proportion" shows up:
1. Ratio of female-to-male honeybee numbers in a beehive.
2. Ratio of diameters of successive spirals in the shell of Nautilus (a marine mollusk).
3. Ratio of diameters of successive spirals in a Sunflower seed pattern.
4. Pinecone petals, leaf arrangements on plant stalks, insect segmentation,...
In a human body too, one sees Φ in the ratio of the lengths of
1. Head-to-toe and navel-to-toe.
2. Shoulder-to-fingertips and elbow-to-fingertips.
3. Hip-to-floor and knee-to-floor.
4. Finger joints, toes, spinal divisions,...so on and so forth.

But the novel is about a fictional story (barring "all descriptions of artwork, architecture, documents, and secret rituals"), and who chases facts in a fiction! However, there are numerous places where such evidences are talked about (just do a google on "golden ratio in nature", or click here for one such example). Leonardo Da Vinci was apparently so impressed that he himself discovered the list of Φ hidden within the human body.

While the list is certainly impressive, the claim is at best only approximately true, contrary to the exactitude typically associated with a mathematical number. Nature does not follow Number Theory, and natural systems always have wide variations around some mean value. Take the human body for example. No two human beings have identical lengths of the same part of their body, and these lengths are in fact continuously varying parameters of human anatomy (in evolutionary biology this is an example of a quantitative phenotype). The ratios mentioned in the above list can only approximate the value of Φ. But if this is true, then what about, say, the ratio of shoulder-to-elbow and elbow-to-fingertips? Suppose this ratio is some μ. If one looks hard enough, I am sure μ can be found to roughly hold in several other parts of the human body. In other words, Φ is about as fundamental to human anatomy as any μ, σ or ω.

Now take the first item in the list: the number of female and male honeybees in a beehive. There is interesting biology involved with this example, and Fibonacci himself supposedly noticed it. If an egg laid by a female bee hatches without being fertilized by a male, it produces a male bee (by a process known as parthenogenesis). On the other hand, if the egg is fertilized by a male bee, it hatches a female. Let us track the genealogy of a male bee. It has a single parent (1 female bee and 0 male bee), which has two parents (1 female and 1 male). The male again has a single female parent, and the female has male+female parents (2 females and 1 male). Each of the two females has a male+female parent pair and the male has one female parent (3 females and 2 males). Three females again have male+female parents each, and the two males have single female parent each (5 females and 3 males), ad infinitum. Going far back into the ancestry, we recover the golden ratio from the female-to-male numbers. Mathematically interesting, but again biologically approximate! In all beehives there are a large variation in the actual male-female ratios due to any number of stochastic factors. What about the other entries in the list? Same argument holds without exception: shell of a Nautilus or seed patterning in a Sunflower never describes perfect spirals, and their diameter ratios give only approximate Φ. These rough spirals are no mere coincidence though, but give evidence to Mother Nature's efficient design, for instance to optimize seed spacing in the Sunflower (see picture).

While writing this post I came across Devlin's Angle, and most of what I said here, he said only better. It is an interesting coincidence that we both start our piece with the same novel, which speaks of its popularity (thanks partly to the transatlantic copyright controversy that made national headlines not so long ago). My advice on if you should read the book: if you like thrillers that are thicker than your pinkie, get it from your local library (as I did) or get your dumb friend to buy it (I do not have dumb friends), but do not waste your dough.