The Fibonacci sequence in Nature

....named after Leonardo of Pisa, a thirteenth century mathematician, known as Fibonacci. His book Liber Abaci written in 1202 introduced the sequence to Western European mathematics, although the sequence may have been previously described in Indian mathematics.

Exponential Fibonacci numbers appear in biological settings, such as branching in trees, arrangement of leaves on a stem, the fruit spouts of pineapples, the flowering of artichoke, an uncurling fern, the arrangement of a pine cone, etc. They are used in the analysis of financial markets, in strategies such as Fibonacci retracement, and are used in computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure.

The simple recursion of Fibonacci numbers has also inspired a family of recursive graphs called Fibonacci cubes for interconnecting parallel and distributed systems.

This is how it can conjugated:

A tiling with squares whose sides are successive Fibonacci numbers in length:


The golden spiral: A Fibonacci spiral created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling:


Cutaway of a nautilus shell showing the chambers arranged in an approximately logarithmic spiral:

Wikipedia

In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:

Each term in this sequence is simply the sum of the two preceding terms and sequence continues infinitely...

1+1=2
1+2=3

2+3=5
3+5=8

But this sequence is not all that important; rather, it is the quotient of the adjacent terms that possesses an amazing proportion, roughly 1.618, or its inverse 0.618.

This proportion is known by many names: the golden ratio, the golden mean, PHI and the divine proportion, among others.
It is found by dividing one number in the series by the number that follows it. For example: 8/13 = 0.6153, and 55/89 = 0.6179.

So, why is this number so important?
Well, almost everything has dimensional properties that adhere to the ratio of 1.618, so it seems to have a fundamental function for the building blocks of nature. Take honeybees, for example. If you divide the female bees by the male bees in any given hive, you will get 1.618.

This same ratio can be seen in relationships between different components throughout nature. The same goes with a human arm that divides into the PHI sequence from the shoulder - through the elbow than hand - right to the finger tips! Sunflowers, which have opposing spirals of seeds, have a 1.618 ratio between the diameters of each rotation:

Wikipedia

Romanesco cauliflower:

Wikipedia

Ferns:


Pine cones:


And even cyclones as this one, evolving offshore Iceland:

Wikipedia

The arms of spiral galaxies too often have the shape of a logarithmic spiral, here the Whirlpool Galaxy:

Wikipedia

A section of the Mandelbrot set following a logarithmic spiral:

Wikipedia

Do yourself a favour... Even if you have little time, go and watch this video on You Tube from filmmaker Cristobal Vila: "Nature by Numbers,"... It's truly beautiful and fantastic!!

Compiled by Noushka