Why an obscure mathematical formula from Middle Ages is critical to the world around us

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.