The divine proportion of pineapples

The Golden Ratio
June 27, 2003

Take a breadstick, break it into two pieces. If the ratio of length of the whole to the longer piece equals the ratio of the longer to the shorter, then the breadstick has been divided into pieces whose lengths are in the golden ratio. It is the ratio of the larger fragment to the smaller. It is a never-ending decimal, equal to approximately 1.61803 (therefore roughly in the proportion of 8:5) that hides throughout the mathematical and artistic literature under the pseudonyms of the golden mean, the divine proportion or the golden section. To the mathematician it is not pi but phi.

All manner of myths have developed about the ubiquitous presence of the golden ratio in the architectural constructions of the ancient Egyptians, the Babylonians and the Greeks. It seems to have supplied an aesthetically pleasing proportion that provides a basis for the design of everything rectangular, from Parthenons to TV screens. Careful draughtsmen such as Leonardo da Vinci and the anonymous creators of the fabulous Islamic tilings seem to have taken this lustrous ratio to heart.

Neither is the story one of human invention alone. The natural world evolves its spirals in ways that pack most efficiently and thus unconsciously seek out this special ratio from all possibilities. Evolution is often golden. Already one is being tempted into dangerous territory. Is there a magic number that can tell you everything you want to know about the world without the labour of observing and thinking? For thousands of years, there have been many who have thought so. They followed the path of numerology, down which they sought to divine the truths of things from the inner meaning of the numbers that describe them.

To the outsider this might sound a lot like modern mathematics. But there is an important difference. Whereas the ancient numerologists sought meaning in the numbers themselves, mathematicians have learnt that the effectiveness of mathematics lies in discovering the relationships between numbers. Thus mathematics is alive with transformations, algorithms, mappings programs, operations, relationships and graphs. All express the deep structures that exist between numbers. Mathematics is usefully thought of as the collection of all possible patterns - there is thus no mystery why a world that contains any order at all should admit a mathematical description. The mystery about our world is that such simple patterns are so far reaching and useful in their deployment.

The golden ratio is one of those patterns. It is simple in definition but mysteriously ubiquitous in its presence across superficially unrelated aspects of the world. Mario Livio has provided us with a well-balanced multicultural exploration of the "discovery" of the golden ratio from the ancient Greeks to the present. His book gently explains many of the intriguing arithmetical properties of phi while tempering the story with an education in parts of the history of art and architecture.

The author is an astronomer at the Hubble Space Telescope Science Center in the US but has maintained a long "amateur" interest in the golden ratio and its manifestations across the whole of art and science. He shows all sorts of unusual examples of the golden ratio's hidden presence in works of art around the world, while carefully sifting the sense from the nonsense in the grander claims for the central presence of the golden ratio here, there and everywhere. Livio shows how in situations with many ratios to choose from, it is all too easy to pick out a golden one if you are looking for it. And alas, the golden ratio is not the design plan for the great pyramid after all.

But away from pyramidology, there are lots of simple eyeball tests that you can try for yourself to see if you most prefer the rectangle whose sides are in the golden ratio. Yet the beauty of this subject is that even after all the nonsense has been debunked, there is still a treasure trove of beautiful things to expound. Livio does a great job in catching them all.

From Penrose tilings and pineapples to limericks, and musical composition to quasicrystals, the subjects change in unexpected ways to keep the readers engaged, whether they are looking for mathematics, art or just good stories. There are copious notes and references to the literature and (even more important these days) to the materials out there on the web.

This is a fascinating story that shows how very simple mathematics provides us with an understanding of natural and human creations that we could obtain in no other way. It is ideal for mathematicians looking for fascinating extensions and for students of aesthetic measure wanting to sift the wheat from the chaff: a better story than pi.

John D. Barrow is professor of mathematical sciences and director of the Millennium Mathematics Project, University of Cambridge.

The Golden Ratio

Author - Mario Livio
ISBN - 0 7472 4987 3 and 4988 1
Publisher - Headline
Price - £15.00 and £7.99
Pages - 294

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