Friday, 1 August 2014

the mathematics of nature

In The Hitchhiker’s Guide to the Galaxy, Douglas Adams famously wrote that the answer to “the Ultimate Question of Life, the Universe, and Everything” is 42. He was wrong, of course. The correct answer is 1.618034 (to six decimal places), a number that is usually represented by the Greek letter Φ.

From whence did I dig up this number? The Greeks knew all about what they defined as the problem of dividing a line segment ‘in mean and extreme ratio’, a problem that is illustrated by the following diagram:
In this diagram, if point C is chosen so that the ratio AC/BC is equal to the ratio AB/AC, then both ratios are equal to Φ. This is the answer to that ancient Greek problem and is now most commonly referred to as the ‘golden ratio’. However, although Φ is usually referred to as a ratio, it is, like its cousin π, the ratio of the circumference of a circle to its diameter, irrational. This means that it is impossible to express as the ratio of two whole numbers, which in turn means that its decimal expansion continues to infinity without ever repeating.

It might seem that this an esoteric concept with little or no relevance to the real world, but if I bend the line AB through 90 degrees at C and add two extra lines to produce a rectangle, then things become more interesting. This so-called ‘golden rectangle’ has sides that are in the golden ratio, and it can be subdivided into a square and a smaller golden rectangle, and so on until the resulting square and rectangle are too small to measure. If I then inscribe a quarter-circle inside each square, the result is a very good approximation to a logarithmic spiral:


This is the same spiral that can be seen in the shells of snails and other gastropods:


The golden ratio is inextricably bound up with the geometry of the circle, as the following two diagrams demonstrate:


In the first diagram, an equilateral triangle has been constructed inside a circle. A and B are located at the mid-points of two of the sides, and BC is an extension of the line AB to the perimeter of the circle. The second diagram shows a regular pentagon and three of its possible five diagonals. In both cases, AB/BC = AC/AB = Φ. Note too that if the two missing diagonals are added to the second diagram, the result is a five-pointed star, or pentagram, which also includes the golden ratio in its properties.

In addition to producing geometric representations of the golden ratio, there are several ways to calculate its numerical value. Perhaps the most bizarre and unexpected involves taking two random numbers and constructing an arithmetic series according to the following rule: starting with the two random numbers, each new term is calculated by adding together the two previous numbers in the series. I’ve chosen the numbers 5 and 7 to begin with.
5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, 898, 1453, 2351, 3804, 6155, 9959, …
The next step is to divide each number in the series by the previous number. I’ve split these calculations into two columns, for reasons that will quickly become obvious:


You will notice that this series of calculations starts with one value that is significantly lower than Φ and a second that is significantly higher, but in the next seven iterations both values gradually converge on the number that I quoted at the start of this account. However, this process can be continued ad infinitum, bearing in mind that only to the extent that the decimal places are the same in both columns can the results be regarded as accurate.

This process may seem no more than an intellectual curiosity, but there is one version of the arithmetic series constructed above that has wide significance in nature. That is the so-called Fibonacci series, named after Leonardo of Pisa (c. 1170–1250), which starts with the digits 0 and 1:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
The special significance of Fibonacci numbers can be glimpsed in the following photograph:


This is the central part of a sunflower. The seeds have arranged themselves in radiating spirals, with one set of spirals curving in one direction and a second set curving in the opposite direction. The surprising thing is that the number of spirals in each direction in any sunflower is always two consecutive Fibonacci numbers. Pine cones and pineapples are also constructed as two sets of opposing spirals, and it turns out that this happens because it produces the optimum packing arrangement in three-dimensional space.

Pine cones also display the double spiral.

Fibonacci numbers reappear in the petals of flowers. For example, lilies have three petals, buttercups five and daisies twenty-one. However, this is not an infallible rule: hydrangeas have four, six or ten petals, and other nonstandard examples include poppies and primroses.

The golden rectangle is often touted as offering the most aesthetically pleasing dimensions for a painting, and Renaissance artists such as Michelangelo, Leonardo da Vinci and Botticelli employed it extensively. The logo of Japanese carmaker Toyota can be enclosed within a golden rectangle.

The golden ratio has also been spotted in famous buildings such as the Taj Mahal, the Parthenon and the Great Pyramid of Khufu, although given that in the latter two cases significant parts of the original building are now missing, some wishful thinking may be involved. People who have identified the logarithmic spiral in the arms of spiral galaxies and the circulation of tropical cyclones may be guilty of the same offence.

That was my first impression. After all, attempts to see the golden ratio in human physiognomy seem to me to be a step too far. However, if the natural path of a seed in a flowerhead is outwards along a logarithmic spiral, and calcium carbonate molecules behave in the same way in the shell of a snail, then why shouldn’t atoms in a spiral galaxy follow the same rules?

And it is quite reassuring to think that everything is governed by a single number.

4 comments:

  1. If there is a God, he must be a mathematician

    ReplyDelete
    Replies
    1. …and a brilliant mathematician too, n'est pas?

      Delete
  2. There are mathematical facts about the golden ratio that are truly interesting. However there is lot of myth also. For instance, the spiral seen in gasteropods has nothing to do with the golden spiral. Same for the nautilus.
    Take a ruler and check your facts ;~)

    ReplyDelete
    Replies
    1. You may say so. I’m not convinced. You obviously failed to notice the word ‘approximation’.

      Delete

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