choose: #### Fibonacci and the Golden Ratio

When a mathematician is asked where mathematics is reflected in the finest way in nature and art, his answer will be: in the Fibonacci series and the Golden Ratio.

What is the Fibonacci series?
The numbers 1, 1, 2, 3, 5, 8,13, 21, 34, 55, 89, 144, .... where each number is the sum of the two previous numbers. The ratio between the last and the second to last number approaches (quickly) the Golden Ratio number $$\varphi$$. ($$\varphi$$ = 1,618 rounded off; see grafh approaching $$\varphi$$. $$\varphi$$ = $$\frac{1+\sqrt{5}}{2}$$ see backgrounds.)

First: nature.

It is remarkable to see how often the Fibonacci series numbers occur in nature. Looking at the sunflowers seed pattern one discovers clockwise and counter-clockwise spirals. The number of spirals are always Fibonacci numbers. And that is also the case with several plants:      The explanation of this phenomenon is that nature is extremely efficient in field division. The position of leaves and seeds enable them to receive a maximum amount of light and water.
Without getting too deep into mathematics it is clear how nature managed evolutionarily. See how growing squares fill the plane: The rectangles grow with Fibonacci numbers. Each new rectangle is positioned against the last one, resulting in a rotary movement. By drawing a quarter circle in each square makes the movement even clearer, the spiral becomes visible: The rectangle gets Golden Ratio in the long run. This rectangle has the quality that the ratio of the width and the length is the same as the ratio between the length and added up the width and the length.
Using the letters above: B/A=$$($$A+B$$)$$/B. How $$\varphi$$ is calculated: see backgrounds.

The pentagram has the same characteristics:  Here also emerges $$\varphi$$: A/B = $$($$C+D$$)$$/D = D/C = $$\varphi$$. For the calculation: see backgrounds

The fivepointed star surrounded by a regular pentagon always captured the imagination. Because of its fine geometrical characteristics several powers (positive and negative) were linked to pentagram in the past. This began with the followers of Pythagoras, and also in the Cathar world (12th / 13th century) the symbol was used. And also in the present day it is widely used: the stars of the flags of the EU or the US, the symbol of the freemasonry and in many other logos the pentagram emerges.

Spanish graphic designer Cristóbal Vila made a fine video about the Fibonacci numbers. Yet there is one thing: he shows also images of the Nautilus shell, this is an example of exponential growth, the Fibonacci numbers are irrelevant here, see also Examples of the Golden Ratio? and Logarithms):

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