LE NOMBRE dor

THE GOLDEN RATIOLyce Louis BarthouMay 2012

by Catherine Le Treut

A gEomEtriC CUTEUCLID (Greek, -300 ) explains a construction for cutting a line in extreme and mean ratio"

CALCULATE THE GOLDEN RATIOx length of the medium segmenty length of the small segment

The name of Greek sculptor Phidiasphi is used to symbolize the golden ratio x + yyx

3

THE GOLDEN RATIO IN HISTORYIn 1509, FRA LUCA PACIOLI, Italian mathematician and Franciscan friar talk about it in his book:De divina proportione Leonardo da Vinci illustrate the book and name the ratio: sectia aureaIn 1596, Johannes KEPLER (German astronomer) talk about it again as a jewel of geometry.

During 19th century, the philosopher Adolf ZEIZING name this ratio the golden section and look for finding it in Roman and Greek temples (the Parthenon, ) , Greek sculptures, European cathedrals, .. In 1932, Prince Matila Ghyka, Romanian diplomat et mathematician, call it the golden ratio and try to find it everywhere

THE GOLDEN RECTANGLEABCD is a square, I middle of [AB] , E a point of the half-line [IB) verifying IC=IE.

It can be shown with the Pythagorean theorem :5The gOLden spiralSo, we can build series of golden rectangles and therefore plot a golden spiral. That spiral can be found in some shells.

6The golden triangles

7THE REGULAR PENTAGON

8

THE PENTAGRAMThe addition of a regular pentagon and a star pentagon gives the pentagram. In the center of this pentagram, we can build and other pentagram and so on...This figure was the emblem of the Pythagoreans.

9A continued fractionThe most irrational amongst irrational numbers .

10The golden ratio in artMany people looked for the golden ratio into art worksIs the Parthenon design really based upon the golden ratio ? Drawings recommended by those who sustain that theory seem complicated

11

The golden ratio in artSeurats painting

Based upon the golden ratio?Or by dividing in two, then in two, the in two . i.e. in 12The golden ratio in artSome artits have used the golden ratio in plentyof their art works

Jacques Villon, french cubist painter

The famous architect Le Corbusier when he creates the cit Radieuse in Marseille13The golden ratio in artWhat about in the comic books ?The golden points in a rectangle are defined by dividing each side according to the golden ratio.

Please check that these plots define a golden pointCould You find a golden point here?The crab with the golden claws. Herg14FIBONNACCI sequenceLeonardo Fibonnacci (12e/13e century) Italian Mathematician, also named Leonardo PisanoHe has worked on the following sequence of natural numbersThe first two equal 1A term can be obtained with adding the two immediately before

The sequence of the ratios of two following terms tends towards the golden ratio

1 1 2 3 5 8 13 21 34 55 89 144 .15FIBONNACCIs rabbitsFibonnacci has used a population of rabbits as an example for his sequence. For each step:A couple of baby rabbits becomes a couple of adult rabbitsA couple of adults doesnt die, and gives birth to a couple of babies

16The golden ratio in nature

Geospace modelling

The golden angle divides a circle into two arcs whose ratio is . As the most irrational number, its the one which allows the optimal filling of space for the formation of plants, pine cones, the position of leaves around a stem.

If you count the number of spirals in a sunflower or a pine cone, in one direction and then in the other, you will find two following terms of the Fibonnaccis sequence.

17

Golden ratio at the cathedrals timeYou can find again the terms of the Fibonnaccis sequenceCathedrals or abbeys builders used length measures inspired by the proportions of the human body.These measures could be found on the builders stick, named la QUINE.One ligne is 0,2247 cm (the diameter of a barley grain)Paume 34 lignes 7,64cmPalme 55 lignes 12,36 cmEmpan 89 lignes 20,00 cmPied 144 lignes 32,36 cmCoude 233 lignes 52,36 cm

18

A golden man?CONCLUSION ?Some people say that the ideal proportions for a man (or a woman) follow a golden ratio :

So try!

ST : heightSN : distance ground belly buttonNT : distance belly button-top of the head

GTN

19HeimildirPWR: THE GOLDEN RATIOhttp://www.europeansharedtreasure.eu/detail.php?id_project_base=2011-1-FR1-COM06-24441