Presentation:
The Golden Mean, or Golden Ratio, or Golden Number is one of these mysterious irrational numbers, like e or (Pi). It is often called or (Phi). is said to be the divine proportion, the ratio of beauty. Indeed, it has been found in some Nature constructions, later re-used in architecture & paintings.

My research:
The value of
The positive result of the equation "X²=X+1" gives value:

= (1+5)/2 = 1,61803398874989484820...
'= (1-5)/2 = -0,61803398874989484820...

NB: you can find the value of the Golden Number with 2000 places here.

is also the limit of the ratio of Fibonacci series numbers:
Lim [n] ( U[n+1]/U[n]) =
where U is defined as follow:

Fibonacci serie
U[n+2]=U[n+1]+U[n]
U[0] = 0, U[n]=1
U[n] values are: 0, 1 , 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc.

Mathematical properties of
In both Arithmetics & Geometry, has many properties that we cannot develop here.
The basic knowledge to keep in mind with the Golden Number is:
    ²=+1.
     and ' are very close as far as their decimal part is identical and they have the following properties: .'=-1 and +'=1.
    Geometrically, is defined as follow:

Euclide (~300BC) called this geometrical drawing
"to divide a line in mean & extreme ratio"

    It is important to know that is particularly present in the geometry of Pentagon (2D), Pentagram (2D) and Dodecahedron (3D):

Pentagon	Pentagram	Dodecahedron

The Golden Number everywhere
Here are some famous examples of use of in the Nature:
- The Nautilus shell (Nautilus pompilius) grows larger on each spiral by ,
- The sunflower has 55 clockwise spirals overlaid on either 34 or 89 counterclockwise spirals, a proportion,
- The drone genealogy follow the Fibonacci serie (so is linked to ),
- etc.

has also been used by artists (painters, sculpters...) who were trying to rationalize aesthetic and understand what make us think whether a shape is nice / harmonious / pleasant or not. It has been used at Keops in Egypt, at the Parthénon in Athens, at Epidaure (Greece), etc.

Leonardo Da Vinci's Corpo Humano
is a good illustration of artists in the quest of rationalizing beauty

How to use quickly the Golden Number?
When you take the first ratio coming from Fibonacci serie, for instance 5/3=1.66..., 8/5=1.6 and 13/8=1.625, you have an approximation of for not precise works (for instance painting). Squaring of 3, 5 or 8 are easy to draw when preparing material for painting, and the previous ratio can help you quickly use "divine proportions".

Another solution to quickly use Golden Mean is to use a Golden Compass, as provided by Robert Losson (craftshiman).