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Monday, 16 April 2012

Phi and spirals

This gets a bit more complicated, but it is still possible to play with this, without too much brain work—it's all about perception.  Still, we decided to leave the mathematical bits out of a book where they might discourage some readers from the main message, which is that poking around nature is fascinating.

The golden mean, or golden ratio, was known to the ancient Greeks. This is a number, represented by the Greek letter "phi", which can be defined as a number which, when you take one away, is its own reciprocal. In algebra, if you understand that, (phi-1) = 1/phi.

A few quick trials on a calculator will give you a value of about 1.618 for phi: if you try, you may even get a better value than this. A line can be divided to make two segments, A and B. If the division is at the "golden ratio" point, then A is to B as A + B is to A. In mathematical notation, A:B = (A+B):A. The Greeks used the golden ratio quite a lot in architecture and art, and so do we: modern televisions and monitors have an 8:5 ratio, which is 1.6, very close to phi.

The curious thing is that phi can be used to draw a spiral. Not only that, some of the most beautiful spirals in nature seem to be based on phi as well. Snails and sunflowers cannot do complicated arithmetic, but the "phi" pattern is there for us to discover in the snail's shell, or the sunflower's seeds.

Here is how to draw an approximation of a logarithmic spiral with just a straight edge and a pair of compasses. Draw the lines first, and then add a series of quarter circles as shown.

Start by drawing a large rectangle with a phi ratio for the two sides: try 18 cm by 29.1 cm, which just fits on an A4 page.

Draw a square (18 cm x 18 cm) at one end, and you will produce a smaller golden ratio rectangle at the other end.

Turn the paper 90 degrees, and make another square. Keep turning the same way, drawing smaller and smaller squares in the same way, and then use a pair of compasses to join the matching corners as shown in the sketch.

Each curve is part of a circle, but the result is very like a proper logarithmic spiral, like the one you can see in a nautilus shell, or in a fossil ammonite, like the one above, which I bought in Morocco, last May.

(A tip for would-be fossil buyers in Morocco: these ammonites are very common, cheap as chips and hard to fake, so they will typically be the Real Thing. Buy one of these as soon as you can and get used to its feel—and the sound it makes when you tap it with a fingernail.  Then, when somebody tries to sell you a trilobite, you are more likely to notice the differences between stone and resin castings.  Sadly, most of the trilobites that you will be offered, especially the fancy ones, are fakes.  There are a couple of photos of a poor-quality fake in my blog entry Simulating a fossil part 1.)
Some amusements

* The Fibonacci series (1, 1, 2, 3, 5, 8, 13, 21 . . . covered in the last entry) can be used to generate phi. Just divide each successive term of the series by the term before it to get a set of numbers that converge on phi. Try any other "cumulative series" beginning with any two numbers, and see what happens. (Two examples to investigate: Lucas numbers and Q numbers.)

* Search around for information on the "golden ratio" or "golden mean", also called "tau". You could also look for information about a Greek artist called Pheidias, who was interested in phi.

* The logarithmic spiral is the only spiral which does not change its shape as it grows: does this explain why the spiral is so common in nature? Can you see a link with fractals?

* Can you use a cumulative series, Fibonacci or your own, to draw a spiral?

Just to prove that the age of the book isn't quite over, here are two references: (Sir) Theodore Andrea Cook, The Curves of Life. New York: Dover, 1979 and Martin Gardner, Ambidextrous Universe, 2nd edition. Ringwood: Penguin Books Australia, 1982.)

I pick up this theme again in Handedness in shells.

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This blog covers quite a few different things, so I tag each post. I also blog about history, and I am currently writing a series of books called Not your usual... and the first two have been accepted by Five Mile Press, The offcuts appear here with the tag Not Your Usual... . For a taste of Australian tall tales, try the tags Speewah or Crooked Mick.   For a miscellany of oddities, try the tag temporary obsessions. And language us covered under the tags Descants and Curiosities, while stuff about small life is under Wee beasties.

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