## Phi

One problem mathematicians constantly struggle with is that there are just not enough letters in the world: we long ago exhausted the entire Roman and Greek alphabets (and even some of the Hebrew one), and as a result many letters are used in a bewildering number of different contexts. Well, I will straight away put your troubled minds to rest by stating that, continuing my occasional series on important mathematical constants (am I allowed to call two posts a series?), when I say “phi”($\phi$) I mean the number$\phi ,$otherwise known as the Golden Ratio, or the Divine Proportion. But first, here are some numbers:

$1,1,2,3,5,8,13,21,34,55,89,144,...$

This is the Fibonacci sequence, and it pops up all over the place.  It is generated by a very simple rule, which I won’t reveal in case you haven’t seen it before (try to work it out).  Now, before I start enthusing about the prevalence of the Fibonacci sequence and the golden ratio in nature, first a disclaimer: one problem with looking for appearances of these sort of things is that people can end up getting a bit obsessed and start seeing them everywhere.  I don’t doubt that some of the claims are little more than a mixture of conspiracy theory, coincidence and approximation; and I have tried to filter out the more wacky theories.  Some things are clearly more than coincidence though!

The Fibonacci sequence is behind many patterns in nature.  Take for example the arrangement of leaves on plant stems, known as “phyllotaxis”.  Many plants have leaves which grow in a spiral pattern: they will be the same lengthwise distance apart, but the direction in which they grow will rotate around the stem.  Look at the leaves on the left in this diagram: between the direction of one and the next there is a rotation of  half of a full circle.  In other plants there might be different rotations, for example in beech trees it is$\frac{1}{3}$of a full rotation, in oak trees$\frac{2}{5},$in poplar and pear trees$\frac{3}{8},$and in willow and almond$\frac{5}{13}.$  Do you see the pattern?  The fraction is almost always made up of two numbers which are alternate entries in the Fibonacci sequence.

A coneflower seedhead: the numbers of spirals in each direction are Fibonacci numbers

Spirals appear all over the place in botany, and very often they are somehow connected with the Fibonacci sequence.  If you look carefully at the pattern the segments on a pine cone or pineapple make, you will be able to discern two sets of spirals heading in different directions – the numbers of spirals are Fibonacci numbers.  Similarly the seedheads of sunflowers and coneflowers, the florets of broccoli and cauliflower, the spines of cacti, and many more.

Fibonacci spiral: not the shape of snail shells

And plants are not the only organisms whose growth is determined by the sequence.   The diagram on the right is known as the Fibonacci spiral: by arranging squares with area equal to Fibonacci numbers in a certain way (note that the length of the side of a given square is the sum of the lengths of the two preceding ones), we can draw a very pleasing curve through the vertices.  This immediately puts one in mind of a snail shell, or perhaps an arm of a spiral galaxy.  (In reality both mollusc shells and galaxies are not quite Fibonacci spirals, but I won’t pass up an excuse to include a nice picture).

I haven’t mentioned$\phi$yet, but it is intimately connected to the Fibonacci sequence.  If we take the ratios between successive members of the sequence we get another sequence as follows:

$1,2,1.5,1.666..,1.6,1.625,1.6153...$

These numbers get closer and closer to$\phi=1.6180339887...$As with$\pi,$this number is irrational, so its decimal expansion goes on forever.  But in the same way that$\pi$has a pleasingly simple geometric interpretation (the ratio of a circle’s circumference to its diameter), so too does$\phi.$If you divide a straight line into two parts, such that the ratio of the length of the long part to the length of the small part is the same as the ratio of the length of the whole line to the long part, then this ratio is$\phi.$  As well as having fascinating mathematical properties, this ratio is ubiquitous in nature, and – probably as a result of this – we as a species find it very aesthetically pleasing.

The golden ratio

It is used prolifically in art, especially in the form of the golden rectangle, which has long side in a ratio of$\phi$to its short side (the spiral diagram above is contained in a golden rectangle; note that whenever you take a square away from such a rectangle you are left with another golden rectangle).  The shape appears in various places on the West facade of Notre Dame, and the UN building is designed in the shape of three golden rectangles stacked on top of each other – one for each 10 floors.  Perhaps the most famous use of the rectangle in architecture, however, is in the Parthenon in Athens.

The Parthenon: wishful thinking?

And this is a good point at which to reiterate what I said about people tending to find whatever they look for!  The picture on the left is a typical example of this.  While the front facade of the Parthenon is indisputably contained in a perfect golden rectangle, the other claims that seem to be being made about the appearance of the ratio in its design are rather more difficult to accept, as the inner red lines don’t actually seem to be aligned with any major features…

Vitruvian Man

But I’m afraid I seem to be undermining my own thesis with all this talk of wishful thinking and conspiracy!  And my next topic will probably not help either.  Yes,$\phi$also often appears in the works of Leonardo da Vinci.  And disregarding his unfortunate modern associations with half-baked conspiracy theories, this is quite significant: he is famed for his precise anatomical drawing, and it is a fact that$\phi$appears in our own bodies in a huge number of places.  Golden ratios abound in da Vinci’s famous “Vitruvian Man”, for example.  These range from golden rectangles that have actually been partially explicitly drawn in – such as that comprising the figure’s torso – to more hidden appearances.  The circle in the drawing takes as its radius the length which forms a golden ratio with the side of the square.  And the length from the bottom of the figure’s feet to its navel is in a golden ratio with that from the navel to the top of the head.

Mona Lisa’s head fits perfectly inside a golden rectangle, and the distance from the bottom of her face to her eyes is in a golden ratio with that from her eyes to the top of her face.  And this is not just artistic license.  To some degree almost every dimension of our faces can be seen to conform in some way to the golden ratio,  even down to the shape of our teeth (our two front incisors approximate a golden rectangle, and the ratio of one front incisor to the next smallest tooth is roughly $\phi.)$It has even been suggested that a person’s perceived attractiveness is partly dependent on how close certain measurements of their facial features are to golden ratios.

I could go on for some time yet, but I think that is enough golden ratios for now.  Clearly$\phi$is a very important number, and as with$\pi,$meditating on why this should be can be rather mind-boggling.  Personally I find that I tend to get entangled in thought-loops:

Why is$\phi$so important?  Because it defines the golden ratio.  What would the golden ratio look like if another number defined it?  Well, then it wouldn’t be the golden ratio.  So what is$\phi,$exactly?  The decimal expansion is just one of endless possible representations, so this is not important.  It certainly does not count anything, as it is irrational, so the number of “things” that it represents is ill-defined.   It just appears to be some kind of mysterious “dimension of the universe”, and it is unlikely that we will ever be able to do better than that.

If you can, I’d like to hear it!  Otherwise it’s probably best not to think about it too much.