More Uxbridge and the life of Phi

I have quoted a few examples from The Uxbridge English Dictionary in previous blogs and I thought I would include a few more choice examples

Lymph – mythical woodland seductress with a sore leg
Cognac – to trick a long-haired Himalayan animal
Vagrant –a  taxi driver’s conversation
Hirsute – what she wears to work
Teacake – a persistent pain in a piece of hardwood
Condemn – how to fool people into giving you money
Decorum – what you do to apples before you cook them

I came across Phi the other day. I am talking about Phi the irrational number. I had heard previously a golden ratio before and read about the perfect ratio 1.618 in the book  “The Da Vinci Code” but not come across it being referred to as Phi. Irrational numbers cannot be written in figures which terminate with zeroes or repeating decimals. So Phi is like pi you can just go on adding numbers to the end ad infinitum. The number is actually 1.6180339887… so Dan Brown actually got it wrong in his book by just stating it was 1.618. Or may be he just got it approximate, if you can do that.

The concept of Phi is ancient; it was apparently named and defined by Euclid in Ancient Greece. It turns out that Phi has some very special properties. Let me tell you about a few that have caught my imagination. Keeping on the Dan Brown theme, in “The Da Vinci Code” he mentions the Fibonacci series 0,1,1,2,3,5,8… where you get the next number in the sequence by adding together the two previous ones. Incidentally this series is named after an Italian mathematician who introduced the series in a book published in 1202 called Liber Abaci (sounds a bit like a old “camp” pianist!). It turns out that if you divide one Fibonacci number by the previous number in the sequence then the result approximates to Phi. Eventually the answer converges as Phi if you could go on for ever.

If you subtract 1 from Phi you will get an irrational number 0.6180339887… and this is also the result if 1 is divided by Phi. Which I find a bit freaky.

Another 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 numbers taken to the limit. It also shows, bearing in mind that Phi is irrational that it is never possible to divide up the line in this way precisely. Any attempt will always be approximate, but maybe near enough.

Phi can be expressed as (1 + sqrt(5))/2 and obviously sqrt(5) is where Phi gets its irrationality from.

There is masses of speculation about the “naturalness” of the golden ratio and its intrinsic appeal to the human eye. Many works of art have the ratio of the width of the picture to the height equal to the Golden ratio, or close to. Widescreen television sets possess the similar characteristics. The computer screen I am using at present is 1280×800 pixels, a ratio of 1.6. More subjective probably is the application of the Golden ratio in works of art and nature. See, for example, http://www.goldennumber.net/ where the concept is applied to design and composition, theology, life (including the dimensions of teeth!) and cosmology, among other things.

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2 Responses to More Uxbridge and the life of Phi

  1. Cath says:

    Maybe Dan Brown rounded up! 🙂

    • seclectic says:

      That would be rational! It might have spoilt the rhythm of his pacey story to have put the number in “in full”. Thanks for taking the time to read my post.

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