Who among us has never gone to a beach and run some sand through their fingers? I certainly have. Did you notice that the grains of sand are all of the same size? Proverbially, two grains of sand are indistinguishable. And so it is with two drops of rain or two snowflakes in the same storm. The same phenomenon can be found in the animate world. Two ants from the same anthill are as similar as the petals on a flower or the seeds of a sunflower. Why? The logic behind it is worth an article of its own. Suffice to say that there are certain common reasons, like some filtering effects in nature, that provide reasons and even an advantage to being identical. How then have all these identical items come about? Through specific processes involved in their creation. Usually, there is just one process for a given item. For ants, the process is birth from an egg. For raindrops, it is vapor condensation, and for grains of sand it is the breakup of a bigger rock.
Let's focus on the sand. Imagine that there was one very large rock that, through continuous breakup into smaller and smaller pieces, eventually becomes a pile of beach sand. Clearly, there is some process that breaks up the rocks. It is not important to me exactly what this process is. Let's just say that it is some combination of temperature change and the rubbing of one rock against another. As with other processes, there is one that is dominant, breaking rocks in the same fashion and proportion into its resulting parts.
Now it's time for some medieval math. A 12th century Italian mathematician in the city of Pisa, Leonardo Fibonacci, defined a simple sequence of numbers as follows: Start with zero and one (0, 1), then generate subsequent numbers by taking the sum of the two previous integers. The first two added together produce 0 + 1 = 1. So the third element of the series is 1, while the fourth is the sum of the second and third elements, i.e. 1 + 1 = 2. Continuing on, we generate 1 + 2 = 3 as the fifth element, 2 + 3 = 5 for the sixth element, then 3 + 5 = 8, and so on. You can easily compute that the Fibonacci series looks like 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc., ad infinitum. [Please note that Fibonacci did not discover this sequence – it was first developed in India – but was the first to present it to a European audience.]
As we continue with the series, an interesting tendency can be observed: The ratio between any two consecutive elements quickly converges to a single value. So if we take three consecutive numbers in the Fibonacci series – A, B, and C – not only do we know that A + B = C, but we also find that the ratios B/A and C/B – which is the same as (A + B) / B – continually gets closer and closer to 1.6180339 etc. So if C were 100, then A is approximately 38.2 and B is 61.8. Therefore the ratio of any two neighboring elements in the Fibonacci sequence [except the first few] is approximately 38.2 to 61.8. This ratio is known as the Golden Ratio (also called, among other names, the Golden Mean). It is present around us everywhere. Leonardo Da Vinci employed it in his measurements of human body (Figure below). It can also be found in the movement of stock market prices, in the structure of bee hives, etc. We can see it in the ancient Parthenon statues of Phidias and in the modern architecture of Le Courbusier. The Golden Ratio is everywhere.
The question that has bothered me for some time was: Why? Why the Golden Ratio? Why it is everywhere? In other words, why do the dimension ratios we find all around us conform to the ratios in the Fibonacci sequence? For what it's worth, here is what I think: Imagine that the huge, solid rock mentioned by me at the beginning of this article is the subject of some single process that breaks it first in two pieces, and then breaks each resulting piece into smaller two pieces and so on, smaller and smaller pieces until all that is left is a pile of sand grains. So what should the ratio be of the two pieces resulting from each step of being broken by the process? Remember that we are talking about one specific and well-defined process, so each act of breaking should result in two pieces with the same ratio, a ratio which is characteristic of that process. Well, it seems that only the Golden Ratio will do the trick for us. So if each piece C breaks into pieces A (the smaller) and B, then B is 61.8% of the C and A is 38.2%. If B breaks again in the same ratio, then the resulting pieces B1 (smaller) and B2 (larger) will be respectively 38.2% and 61.8% of B. Please observe that B2 is 61.8% of B, which itself is 61.8% of C. Therefore B2 will be 61.8% * 61.8% = 38.2% of C. The B2 chip of B is the same size as A. Both are 38.2% of the initial element C. These pieces are identical in size despite being produced by two different branches of the process. One was the result of chipping 38.2% from C (this became A), and the other piece was the result of chipping 61.8% of B (which became B2). Different breaking paths resulted in chips A and B2 ending up with exactly the same size. In diagrammatic form, this is what happens:
38.2 61.8
A + B = C
/ \
B1 + B2 = B
38.2 61.8
Now we come to the main thesis of this paper. It is clear that a breaking process that uses the Golden Ratio (which is also the ratio of adjacent elements in the Fibonacci series) produces elements that have the same size. As the breaking process continues, the next generation will always have only two sizes: the bigger piece (61.8% the size of its parent) and the smaller piece (38.2%). The bigger piece will be the same size as the smaller piece from the previous generation. The uniformity of sizes will be preserved; this appears to be the preferred outcome in nature. Any process that divides elements into subelements whose ratios are the same as two adjacent Fibonacci sequence numbers will lead to the creation of uniform elements, regardless of the number of iterations (number of times the process is applied to the resulting subelements). I have not tried to find a proof that only Fibonacci-conforming processes lead to such uniform outcomes, but my intuition tells me that this is the case. Maybe you, the reader can think of a proof?
One final comment: This effect (where the larger piece of the second generation, taken from the larger piece of the first generation, will be the same size as the smaller piece of the first generation) would also be achieved if the process broke pieces not into Golden Ratio-sized elements, but also ratios of 2:1 or 2:3. Significantly, these numbers are themselves part of the initial Fibonacci sequence, which quickly converge to the Golden Ratio. As just single discrete elements of that infinite sequence, these ratios represent an infinitely small number of Fibonacci elements. Hence processes using these ratios must also be infinitely rare.
Daniel GrynglasSan Jose, Dec. 14, 2015