The Golden Ratio – Mario Livio

This is an interesting subject that has fascinated me for years. The book is not for the mathematically disadvantaged…sorry, you need some background in a minimum of geometry in order to make any sense of it.

What is most interesting is the frequency with which the ratio and its relatives have appeared in history and life.

First, something of a definition: “The first clear definition of what has later become known as the Golden Ratio was given around 300 BC by the founder of geometry as a formalized deductive system, Euclid of Alexandria….In Euclid words: A straight line is said is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.” A graphic from the Wikipedia illustrates:

A line is composed of 2 parts (a) and (b). The ratio of a / b = (a+b) / a. The value of the ratio is the never repeating number 1.6180339887…

All this seems…so what?

It is as Einstein remarked: “The fairest thing we can experience is the mysterious. It is the fundamental emotion which stands at the cradle of true art and science. He who knows it not and can no longer wonder, no longer feel amazement, is as good as dead, s snuffed out candle.”

The golden ratio is often denoted by the Greek letter phi (Φ or φ). The figure of a golden section illustrates the geometric relationship that defines this constant. Expressed algebraically:

 \frac{a+b}{a} = \frac{a}{b} = \varphi\,.

This equation has as its unique positive solution the algebraic irrational number

\varphi = \frac{1+\sqrt{5}}{2}\approx 1.61803\,39887\ldots\,

Some examples:

Some thing strange about all this?  Try this on a calculator: Enter the number 1.6180339887 and hit the “square” button. The result will be 2.6180339887. Enter the number again and find the reciprocal. The resulst is 0.6180339887.  This number has the unique qualities that to find its square, you simply add 1; to find the reciprocal, you simply subtract 1……a tad unusual perhaps?

How about those Fibonacci numbers: defined as a sequence of numbers, starting with 0,1 and generating the next number in the sequence by adding together the last 2 numbers. A short list:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …

These have some interesting characteristics of and by themselves, but one that fits in here is found in the ratio of successive numbers in the sequence:

1/1 = 1.000000
2/1 = 2.000000
3/2 = 1.500000

21/13 = 1.615385
34/21 = 1.619048

610/377 = 1.618037
987/610 = 1.618033

hmmmmmm…look familiar?

How about this: take any 10 Fibonacci numbers in sequence….it is always, always, divisible evenly by the number 11…..want another?….the sum of those 10 number is always the 7th in the sequence multiplied by the number 11…

Still with me?

Now, take a look at the last digit (the units digit) of one of these Fibonacci numbers. That units digit, whatever it might be, will also be repeated in the 60th following number. The last 2 digits , i.e. 01 will repeat every 300 times; the last 3 digits will repeat every 15,000 times and the last 4 digits will repeat every 150,000 times….

Amazing or not?

I leave it to you to either read the book or do your own research to identify the numerous and quite amazing instances where the Golden Ratio appears in architecture, nature, art and engineering….have fun.

But I ask as a final question…..is this all accidental or is it the consequence of some intelligent design of the universe?

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