Alex's Adventures in Numberland - Alex Bellos [120]
It follows that a square of 169 unit squares (13 × 13) can be rearranged to ‘make’ a rectangle of 168 squares (8 × 21). In this case the segments overlap slightly along the middle diagonal.
In the early seventeenth century, the German astronomer Johannes Kepler wrote that: ‘As 5 is to 8, so 8 is to 13, approximately, and as 8 to 13, so 13 is to 21, approximately.’ In other words, he noticed that the ratios of consecutive F-numbers were similar. A century later the Scottish mathematician Robert Simson saw something even more incredible. If you take the ratios of consecutive F-numbers and put them in the sequence:
which is:
or (to three decimal places):
1, 2, 1.5, 1.667, 1.6, 1.625, 1.615, 1.619, 1.618…
then the values of these terms get closer and closer to phi, the golden ratio.
In other words, the golden ratio is approximated by the ratio of consecutive Fibonacci numbers, with the approximation increasing in accuracy further down the sequence.
Now let’s continue with this line of thought and consider a Fibonacci-like sequence, starting with two random numbers, and then adding consecutive terms to continue the sequence. So, just say we start with 4 and 10, the following term will be 14 and the one after that 24. Our example gives us:
4, 10, 14, 24, 38, 62, 100, 162, 262, 424…
Look at the ratios of consecutive terms:
The Fibonacci recurrence algorithm of adding two consecutive terms in a sequence to make the next one is so powerful that whatever two numbers you start with, the ratio of consecutive terms always converges to phi. I find this a totally enthralling mathematical phenomenon.
The ubiquity of Fibonacci numbers in nature means that phi is also ever present in the world. Which brings us back to the retired dentist, Eddy Levin. Early in his career he spent a lot of time making false teeth, which he found a very frustrating job because no matter how he arranged the teeth he could not make a person’s smile look right. ‘I sweated blood and tears,’ he said. ‘Whatever I did the teeth looked artificial.’ But at around that time Levin started attending a maths and spirituality class, where he learned about phi. Levin was made aware of Pacioli’s The Divine Proportion and was inspired. What if phi, which Pacioli claimed revealed true beauty, also held the secret of divine dentures? ‘It was a Eureka moment,’ he said. It was 2 a.m. and he rushed to his study. ‘I spent the rest of the night measuring teeth.’
Levin scoured photographs and discovered that in the most attractive sets of teeth, the big top front tooth (the central incisor) was wider than the one next to it (the lateral incisor) by a factor of phi. The lateral incisor was also wider than the adjacent tooth (the canine) by a factor of phi. And the canine was wider than one next to it (the first premolar) by a factor of phi. Levin wasn’t measuring the size of actual teeth, but the size of teeth in pictures when taken head-on. Still, he felt like he had made an historic discovery: the beauty of a perfect smile was prescribed by phi.
‘I was very excited,’ remembered Levin. At work, he mentioned his findings to colleagues, but they dismissed him as an oddball. He continued to develop his ideas nonetheless, and in 1978 he published an article expounding them in the Journal of Prosthetic Dentistry. ‘From then, people got interested in it,’ he said. ‘Now there is not a lecture that is given on [dental] aesthetics that doesn’t include a section on the golden proportion.’ Levin was using phi so much in his work that in the early 1980s he asked an engineer to design him an instrument that could tell him if two teeth were in the golden proportion. The result was the three-pronged Golden Mean Gauge. He still sells them to dentists around the world.
I couldn’t tell if Levin’s own teeth were in the golden proportion, although there was certainly a fair amount of gold in them. Levin told me his gauge became more than a work tool, and he started to measure objects other than teeth. He found phi in the patterns of flowers, in the spread of branches