I have an abiding fascination for numbers that are found everywhere in nature. The Golden Ratio, the Fibonacci Sequence and Benford’s Law are prominent examples. In this post, I show that not only are these numbers beautiful in themselves, they are also intimately related to each other.

The so-called Golden Ratio (approximately 1.618) gives aesthetically pleasing dimensions for rectangles and has been found in all kinds of places from seashells to knots, the most famous example being the spirals of the beautiful nautilus shell (pictured below).

Then there is the Fibonacci Sequence: 1, 1, 3, 5, 8, 13, 21, 34 and so on, where every number after the first two 1s is obtained by adding the previous two numbers. Again, this intriguing sequence crops up all over in nature, from the arrangement of leaves on plants to the spiral pattern of seeds in the head of a sunflower, and even the sea-bubble formations formed by humpback whales (below)

Given that both the Golden Ratio and Fibonacci sequence are ubiquitous in nature, one might suspect that they are somehow related, and and they are! In particular, the ratio of successive terms in a Fibonacci sequence tends toward the Golden Ratio. To illustrate this, let me expand the Fibonacci sequence a bit further to:

1,1,2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 …

First take 5 divided by 3, which gives 1.667. Let’s say we skip three numbers and take 34 divided by 21, giving 1.619. Skip another three numbers and divide 233 by 144, giving 1.61806. And so on. See the direction we’re going? In effect, we are approaching the Golden Ratio as we go further out into the Fibonacci Sequence.

That is the first connection. The second connection is more subtle. It has to do with another number pattern known as Benford’s Law (after the American physicist, Frank Benford).

Benford’s Law

Like the Fibonacci Sequence, Benford’s Law is hidden in many daily phenomena that we don’t even notice. For example, extract a ragbag of numbers from a newspaper (any newspaper will do) and count the distribution of those numbers. You will find that close to 30 percent of the numbers will start with a 1, 18 percent will start with a 2, right down to just 4.6 percent starting with the number 9.

To give a more visual picture of Benford’s Law, consider the first digit of populations in all US counties. (the source for this data is an official COVID-19 report). Count the numbers of counties with populations whose first digit is 1; do the same for counties whose populations whose first digit is 2, and so on. So, if the population is 1,458,935 you take the 1, and if the population is 2,385,790, you take 2 and so on until the digit 9. The following chart plots the frequency distribution of digits 1 to 9 being the first digit of the populations. Amazingly, 1 occurs about 30% of the time, 2 occurs about 17% of the time, and the pattern continues in a manner that very closely conforms to Benford’s Law (give and take a very small margin of difference).

Even more amazing, Benford’s law applies to many other things like stock prices, the number of friends people have on social media, the height of mountains, the length of rivers, and even the distance between Earth and all known galaxies. In all these cases, not only are smaller leading digits more common but they follow a precise and consistent pattern. Moreover, it doesn’t matter what is the measurement metric – the law works whether it’s miles, kilometers, light years, inches…it all comes back to the long-tailed distribution described by Benford.

Exactly why numbers should behave this way remained a mystery until 1996, when mathematician Theodore Hill of the Georgia Institute of Technology proved that Benford’s Law will emerge when one mixes a hotchpotch of numbers, each following its own statistical law. For example, the height of people tend to follow a bell-shaped curve known as the Gaussian distribution, daily temperatures rise and fall in a wave-like pattern, and the frequency of earthquakes are described by a logarithmic law. Now imagine grabbing random handfuls of data from a hotchpotch of such distributions. Hill proved that as your ragbag of numbers get large, the digits of these numbers tend toward a single universal distribution that is Benford’s Law. In other words, Benford’s Law is the inescapable “distribution of distributions”. How wonderful is that?

And to add icing to the cake, there is a connection between Benford’s Law and the Fibonacci Sequence though that is far from obvious: the digits that make up the Fibonacci Sequence themselves conform to Benford’s Law! You can see this by glancing at the table below showing the frequency of initial digits produced using an online Fibonacci calculator (available at https://r-knott.surrey.ac.uk/Fibonacci/fibmaths.html#section7)

So there we have it: the hidden connections between three of nature’s most ubiquitous number patterns. By way of summary, we can say this: if Benford’s Law doesn’t hold, there would be no Fibonacci Sequences, and no Golden Ratio. The fact that the Fibonacci Sequence appears to be everywhere is because we are dealing with phenomena that are the result of mixing a bunch of random numbers from a bunch of random data sets. The more ragbag the dataset is, the more confident we can be that the digit distribution will obey Benford’s Law., the Fibonacci Sequence and the inevitable Golden Ratio. Mysterious are the ways of nature!