For hundreds of years, artists, architects, musicians, mathematicians, and even biologists have been fascinated by this number, seemingly ubiquitous in all we consider beautiful. But what exactly is this number? Where is it derived from? And what gives it such strange properties, and why does it come up so often in unexpected places?
One of the simplest ways to define the golden ratio was proposed by Euclid in about 300 BC. It spoke about a particular point on a line.
Euclid investigated a special point, dividing the line a+b, into a and b, in such a way that the ratio a:b was exactly the same as a+b:a. Perhaps an easier way to envisage this is with a rectangle, often called the “Golden Rectangle.”
The ratio of the vertical side to the horizontal side of this rectangle is clearly 1:φ (pronounced “phi”). However, if we draw another vertical line to construct a square of side one, we produce another rectangle.
Euclid then defined φ as being the number for which this new rectangle formed on the right was similar to the large rectangle as a whole. More mathematically speaking,
Because this new rectangle on the right is also a golden rectangle, it itself can then be split up into a square and a golden rectangle! This process can go on ad infinitum, as shown in the diagram below.
But how do we find a value for φ? We can use the expression above
then multiply by (φ-1) to get…
-1 from both sides to get a quadratic equation
Βut since this is geometry we eliminate the negative value and are left with just 1.61803…
You may have spotted something else funny about φ. We could write our original equation slightly differently by flipping the fractions upside down.
Yes that’s right, φ has the unique and bizarre property that it is exactly one greater than its reciprocal: 1/φ = 0.61803…, but it doesn’t stop there. Using the quadratic equation we used earlier, we can get an expression for φ2…
So φ2=2.61803…! This also gives us an easy way to calculate further powers of φ, by substituting any φ2 for φ+1
There is a fundamental pattern happening here. φ=1 + 1/φ. This can also be written as
In the same way
φ2=φ+1, or φ2=φ1+φ0
In general, you could say that φn=φn-1+φn-2.
So, if we had a series, let’s call it F, of the powers of φ, then Fn=Fn-1 + Fn-2.
In other words, each term is formed by adding the two previous terms together. You may have heard of a sequence like this before, it’s called the Fibonacci series. It normally starts like this…
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 …
The only difference between this and the powers of φ is that we defined the start differently! After the first two terms it follows exactly the same rules for the formation of further terms.
So, if we consider the powers of φ being as being part of the series, then each subsequent term of the series will be φ times greater than the last. This is true by definition, for any number x xn/xn-1=x. This means that in any Fibonacci sequence, no matter what your starting terms, the ratio between one term and the next will gradually close in on φ!
Now we move on to speak about the irrationality of φ. φ is an irrational number, which means that it cannot be expressed as a fraction of two integers, and has an infinite amount of digits that continue without repetition. Another property of irrational numbers is that neither a rational multiplied by an irrational number, or a rational add an irrational number will have a rational result.
However, irrational numbers can be expressed as ‘continued fractions,’ such as the one I am about to show you.
We start off with a simple fact:
Now, wherever we see φ, we can replace it with 1+1/φ
This is actually the simplest continued fraction you can get: the 1’s gradually trailing off to infinity, but this also makes it the ‘most irrational’ number. But how can any number be ‘more’ irrational than another? How irrational a number is depends on how well it can be approximated using fractions. π, for example, can be approximated relatively accurately by 22/7, which is accurate to about 0.001264, but the closest approximation to φ with a denominator under 10 is 13/8, which is only accurate to about 0.00696601125, six times less accurate! This article
It’s for this reason that φ shows up so much in nature. For example, for most plants which grow leaves around a stem, each leaf is rotated exactly 360/φ°, or about 222.5° from the previous one. Nature is taking advantage of the irrationality of φ: this means that no two leaves will ever overlap very much, in fact it is so irrational that it creates the minimum amount of overlap between leaves. This is important because it allows more light to get at each plant so that they can photosynthesize with maximum efficiency.
Again, because it is so irrational, it can be used to aid space efficient packing of seeds. In sunflowers, the seeds are placed in two opposing spirals, one of 89 seeds, and one of 55 seeds, two Fibonacci numbers. These spirals create the Golden Angle we talked about above. This means that two seeds will never line up perfectly with the center of the sunflower, and thus there will always be a small gap for the next seed to be placed into. Other seed bearers, such as pine cones, also have patterns of two Fibonacci spirals.
One of the most famous uses of φ in nature is in the Nautilus shell, and the shells of many other creatures. There is a spiral in their shell patterns as below. This is because of the self-similarity of Golden Rectangles: the Nautilus can keep growing its shell in the same pattern and constantly get bigger, meaning the shape and design is never a limiting factor.
φ appears in many other natural patterns, often growth related, in other areas as well. This diagram shows the many occurrences of φ or its powers in Leonardo Da Vinci’s Vitruvian Man.
Other artists have used φ as well, such as in this piece by George Seurat (top), and this by Piet Mondrian (bottom).
Debussy, the French impressionist composer, is said to have used φ in many of his compositions, with the climax of the piece often coming approximately 61.8% of the way through the entire work, such as in Reflets dans l’Eau, where the harmonic structure also follows the Fibonacci sequence. The recursive, infinite nature of Golden Rectangles and φ seem to be very attractive to artists, so much so that it became known as the “Divine Proportion”, and perhaps that is why φ features so heavily.
Many have also used φ, with varying success, to try to predict the stock market, using techniques such as Fibonacci Retracement, or predicting population growth in various species, mostly based on a more complex version of Fibonacci’s original Rabbit Problem.
Whilst so much about this number and its ubiquity is still a mystery, one thing is certainly clear: φ has a profound effect on so much in our world, whether in nature, art, or mathematics, and we will keep discovering more appearances of this enigmatic number for years to come.
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