John McCullagh January 15, 2008

I am returning to a topic touched upon earlier, the significance of certain concepts in mathematics [and nature] and in particular, to the Fibonacci sequence of number …

…  [1,1,2,3,5,8,13,21,34,55 ] which yields by extension the centrally important quotient phi, equivalent (approximately) to 1.618. The inverse, 1 divided by phi (0.6180) is equally interesting.

At school I always had difficulty with concepts like pi (approximately 22/7 or 3.141592 …) and with precisely where its use and significance lay. I didn’t know the Greek alphabet or why these strange letters should be ascribed certain fixed values. Of course we were made to utilise pi to determine the area and/or circumference of a given circle but it was all in the nature of a number game – or worse, a dreaded SUM!    Pi remained a mystery.. as it largely does to this day. Did you know, for example that the sequence of numbers after the decimal point would never reach an end, no matter if all the material and energy in the world could be converted to printed numbers to fill the spaces and all the computing power therein were applied to this task? At least I’m told this, though how it might be known is an even greater mystery to me. And why we should repeatedly apply ourselves to a concept that is clearly so imprecise!

I looked around me then and could see little use for, or examples of perfect circles [with the possible exception of the sun and the moon] in nature – an impossible concept anyway, I was informed in mathematics class, and thought that all of this belonged only in the driest of all locations, school mathematics textbooks!

The man-built environment loved the perfect lines and curves, squares and rectangles beloved of our mathematics tutors but nature was strictly averse to it all.

The physical and the living environment showed signs of randomness instead, the shapes of clouds, coastlines, river banks etc. and there were very few of the classical shapes on display in the living world too. Mathematics wasn’t real!

Then the concept of phi was introduced to us and more sums set. But it was all taught in isolation. Without circles (pretty in themselves) it was even more meaningless! I could understand when a five-pointed star was drawn and it was revealed to me that the places of intersection of the lines joining points divided those lines in phi proportion, because I could measure this with a rule. But could the teachers not have made all this more immediately relevant by having us measure equivalent proportions in our own bodies?

Here are a few examples : Phi is revealed in the length proportionately of…

Your total height divided by the distance from your belly-button to your soles…

The distance from your shoulder to your fingertips divided by elbow-to-fingertips…

Hip to floor divided by knee to floor,  and

In your finger and toe joints.

And what is true for man is true for the other mammals and other animals, insects etc. If God had a plan for life on Earth, phi was central to that plan!

So meaningful is it all, that in another more religious era this number became known as the Divine Proportion.

The great masters in music, science, philosophy, art and a multitude of other pursuits were acutely aware of the ‘Divine Proportion’ or Golden Number (Proportion) and utilised it freely in their works.

Let us return to the Fibonacci sequence! Proceed to add numbers to the following sequence : 1,1,2,3,5,8,13,21,34,55 … Easy, each consecutive number is the sum of the two preceding numbers. Now divide any of these numbers by the one preceding and you yield phi (1.616 .. or a number progressively approaching that) : or by the number following and you yield (0.6180). As we explained in our earlier article, this can convert, in the case for example of sunflower seeds flowering in spirals in the head to the related angle of 137.5˚ { 360˚ (1-Φ) ˚ = 137.5˚ where Φ = phi} which is the only angle that will yield the maximum number of seeds concentrated in the minimum space – and thus facilitate survival of the species in the competitive world of nature. There is a purpose. Christians who dominated Western Thought for millennia called it God’s Purpose.

Did you know that female bees dominate males in the hive in numbers in exact phi proportion? There are other examples. The cephalopod mollusc (and others too of course) has spirals on its shell the ratio of whose diameter to the next adjacent one is in keeping with the Golden Number, or Divine Proportion, if you like. The same can be found in the metre of poetry and the tempo of music: artwork by Michelangelo, Albrecht Durer, Leonardo da Vinci and others demonstrated each artist’s adherence to the Divine Proportion in the layout of their compositions: it is there in the architectural dimensions of the Greek Parthenon and the Pyramids: and in the organisational structures of Mozart’s sonatas, for example and Beethoven’s famous Fifth Symphony as well as in the works of Bartok, Debussy and Schubert.

Meet phi – also expressible as half the square root of 5, minus one – a very interesting number indeed!


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