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The Weekly Scientist on The Divine Proportion

Defintion of the Golden Ratio

Like many final year students, I am wrapping up my university life in the next 5 weeks by tidying up my dissertation and sitting my final set of exams. Having opted for the 4 year undergrad course in maths (known as the MMaths), I was able to avoid the “real” world for an extra year while most of my friends grew up and started earning a living. With the inevitable end in sight, I wondered if I had truly studied up to current research level (which loosely is something the MMath claims) and if there was anything I hadn’t learned about or at least hadn’t had the chance to, in a mathematical context. I remember finishing Dan Brown’s The Da Vinci Code just before starting at Nottingham and wondering if I would learn about the mysterious “Golden Ratio” that is mentioned in Brown’s book. Having nearly finished a masters, I can safely say — I didn’t. This of course could have been due to a bad module choice or a missed lecture. But even if that was the case, it seems preposterous that I had not encountered it in other modules or lectures as it seemed (from what I had heard) that this mysterious number has great applications and can be found everywhere. So what is it?

The Golden Ratio (or Divine Proportion/Golden mean etc) was first discovered by the Ancient Greeks, specifically by Pythagoras or his followers. In modern history it appears in Luca Pacioli’s De divina proportione (On the Divine Proportion), which most noticeably contains an illustration by Leonardo Da Vinci of the human face broken down into divine proportions. Da Vinci’s illustrations for De Divina Proportione have led to speculation that his Mona Lisa would contain it. Extensive research has shown that it doesn’t.

The Golden Ratio was first defined and named by Euclid (the man who gave us graphs the way we know them today). It is irrational — the decimal expansion begins like 1.61803… — and defined as follows: Two numbers, A and B, make the golden ratio if the sum of the two numbers divided by the larger of the two is equal to the larger number divided by the smaller one (see Image 1). A very good approximation to the Golden Ratio is obtained by taking two adjacent Fibonacci numbers (numbers that are in the Fibonacci sequence) and dividing the larger one by the smaller one — except 0. The further along the sequence you choose to pick the adjacent numbers the closerthe approximation. It is this property of the Golden Ratio that has led to it appearing in nature.

(Image 1: Definition of the Golden Ratio)

The Fibonacci sequence is a very “natural” sequence as the next number is defined by adding the previous two (with the first two numbers being 0 and 1). It has been linked to population models (very basic ones), but most successfully studies of bees have shown that the ratio of female bees to male bees in a typical hive is in fact the Golden Ratio. Because the Golden Ratio is effectively a proportion, mathematicians have tried to find this proportion in other aspects; for example the Golden Angle is approximately 137.51°. This is the angle that separates petals around a stem and maximises the exposure of light on the leaves. Another Golden Ratio derivative is the Golden Spiral, which gets further away from its origin by a factor of the Golden Ratio every quarter turn and can be seen most interestingly in snail shells and seashells. The seed arrangement in sunflowers also exhibit these spirals, but not in an obvious way. There are 89 spirals of seeds in a sunflower of which 55 start spiralling leftward and 34 start spiralling rightward and the effect together is a seemingly randomly covered seeded area in the flower; strangely the ratio of left spirals to right spirals is also the Golden Ratio (approximately). The spiral arrangement can be found in other vegetation such as pine cones, pineapples, cauliflowers’ and countless others.

So what does this mean? Is all life and natural things based on this number? Can it be used to predict the future (like financial markets)? And why didn’t I get lectured on it?

The meaning of the number is purely left for the reader to interpret; practically it’s just a number and it is found in nature due to the Fibonacci sequence being an obvious and natural increasing sequence. This could possibly be why we find items that contain the ratio aesthetically pleasing, like the original iPods or a perfect set of teeth. Alex Bellos’ book Numberland outlines how a dentist uses a special device that measures the Golden Ratio to dictate sizes of teeth when he’s reconstructing them. A few examples of where the ratio occurs in nature are given, but there are countless examples where it doesn’t and could never appear. And there is always that old logical thought that if you look for something hard enough in art, architecture or anything else you can make yourself find it, even if it is found in an absurd manner.

Predicting the future? Possibly. There are algorithms’ and strategies that help predict the trend of stocks that use the Golden Ratio and have with great success predicted market moves. According to scientist Stephen Wolfram, “this seashell may hold the secret of stock market behaviour, computers that think and the future of science” — and on that seashell why wasn’t I lectured on it? Who knows, maybe I was distracted by a wave of self-pity induced by a hangover or the Golden Ratio is just a fallacy kept alive by some inner desire in us to make sense of the universe.

As the Great Einstein once said, “Two things are infinite: the universe and human stupidity; I’m not sure about the universe.”

Phil Bowyer

Editor’s Note: After a long hiatus brought about by a furious spell of dissertation deadlines, The Weekly Scientist has at last returned! Hopefully you will have missed our pontificating as much as we have missed doing the pontificating. From all of our writers, I offer a sincere apology for the lack of updates; hopefully, the withdrawal symptoms weren’t too bad. As for what happens next: The Weekly Scientist is intended to run for another few weeks before our last term comes to an end with a movie montage of us all bursting out of examination halls, donning flip-flops and jumping onto the nearest bus heading toward the beach (any beach). Until then, keep reading and commenting!

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2 Comments on this post.
  • sexy love
    17 March 2012 at 11:12
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    first I appreciate your hard work ,but i think golden ratio first uesed by ancient Egyptian when they built pyramids

  • More Uxbridge and the life of Phi | seclectic
    19 April 2012 at 19:22
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    […] way to define Phi is illustrated in the above diagram taken from http://www.impactnottingham.com/2011/05/the-weekly-scientist-on-the-divine-proportion/. But if you think about it this is just another of thinking about the ratio of the Fibonacci […]

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