The Fibonacci Sequence

The Fib sequence, is one of those strange mathematical entities, that appears to reflect itself in nature.  For those that don't know, the Fibonacci sequence is created, using the real numbers, from 1, 1, 2... to infinity.  The numbers are added together in sequence, so that the previous two numbers in the sequence, add up to the next number.
For example, the first few numbers of the sequence are:

1,1,2,3,5,8,13,21,34, 55, 89 etc...

So 1+1 = 2

Then 1 +2= 3

2+3= 5

And so on.

But, for some reason, the ratios between the numbers in this sequence, as the sequence progresses, have some strange properties.

The ratios include, the so called Golden Ratio, of approx 1.618.  This ratio, is used in creating, so called, perfect proportions, such as those found in 5th Century BCE, Greek architecture; or it is used by some plastic surgeons, when they are trying to create perfect facial proportions.  The Fib ratios are also part of nature and are alleged to be found in the patterns of sunflower seed heads, or the chambers in the shell of a Nautilus.

The ratios are also heavily used by stock and Forex (foreign exchange) traders, as the stock market trends have a funny habit of starting or ending, near to key Fibonacci ratios, within their charts and graphs.

This strange sequence has also lead me to ponder the natural logarithms and the number 'e' where e is an irrational constant approximately equal to approx: 2.718281828
Whilst we tend to find the number 10, easier to work with, when trying to count or measure things; this number 'e', keeps cropping up in maths and in nature, in the strangest of places.


I have no idea why; but this absolutely fascinates me, and it is one of the aspects of maths, that I am going to now find out more about.  How it fits into Physics, is, no doubt, going to be surprising and probably very elegant.