this number possesses the unique and curious arithmetical property to be situated with regard to the number 1
so that if we add 1: we raise it to the square; if we deduct 1: we obtain its opposite
is the geometrical progress of reason . It can express himself according to or
( +1) (2+1)
(5+3) ... in the ascending and downward progress from 1, we find
the terms of the series of Fibonacci (in whom(which) every term
is the sum of both precedents) moved by a rank for the multiple
of and the term which
is added to it
= + et 1 /= 1 /( + ) with the term of the series of rank n
Series of Fibonacci 1 1 2 3 5 8 13 21 34 55 89 .... whose relationship between 2 successive terms aims towards especially since n increase 89 / 55 = 1,61818
And many other curiosities which will make say that will be considered as the cosmic number of the perpetuation.
|Did the ancient Egyptians know
the golden section? Is it contained in the proportions of the
pyramid of Khéops?
It is difficult to know it by the measure, its dimensions being a little bit different today from the origin, this building having occasionally served as stony quarry.
| However, we have the testimony of Hérodote
which, with a long stay in Egypt, received some fragments of
knowledge under the shape of this mathematical enigma:
" The surface of every triangular face is the same that the surface of a square sides of which would be equal as high as the pyramid ".
Enigma which we can resolve easily: the pyramid has a square base with the sides = 2, its height is H, a is the apothème or the height of the triangle of a face.
is connected with the number 5 (by its root) but also in the flat or spatial geometrical figures presenting a symmetry of order 5 or multiple of 5 (pentagon, decagon, icosahedron, dodecahedron, triacontahedron)
Construction of the golden section
Two squares attached aside 1 form the
rectangle ABCD of sides 1 and 2. The diagonal AC cuts the side
common of squares in its middle. With this point for centre,
we draw the circle of diameter 1 which cut AC in M and N.
AM = 1 /
How to draw a pentagon with only the (ruler and the compass? It is not evident. Here is a method
|We draw a circle of center O and two diameters perpendiculars. I is the middle of the radius OD. A circle of center I and of radius IA cuts OB in J. A circle of center A and of radius AJ cuts the initial circle in E and F. A, E and F are 3 summits of a pentagon|
Two circles of radius 1 the centres of which beeing separated from the distance 1 / cut themselves according to a rope side of the starry pentagon
|The side of the convex pentagon measures The side of the starry pentagon measures The relationship between both is|
Also for the decagons
The measure towards(as for) the convex decagon is 1 / = - 1
The measure towards(as for) the starry decagon is
And a main rule for pentagons and decagons: if the distance of the centres of beam(shelf) 1 is the side of the convex polygon, the common rope is the side of the other studed polygon, and conversely.
Regular polyhedron consisted of 20 (4 x 5) equilateral triangles
It contains 3 golden rectangles (proportions 1 and) perpendicular (it is a way of building it)
If the confined sphere has a radius of 1, the arète of the icosaèdre is = 1,051
Composed of 12 pentagon, the regular dodecahedron, dual of the icosahedron, wear the golden section.
If 1 is the length of the side, the diameter of the confined sphere is
Rhombic triacontahedre (or zome 5)
Composed of 30 golden rhombuses (relationship between diagonals =,), the rhombic triacontahedron can be created from the icosaèdre or from the dodecahedron.
and the diameter 2 /
Golden number in the nature
|Logarythmic spirals for the nautile and the flower of the sunflower|