THE GOLDEN RATIO

Lycée 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 Phidias

phi is used to

symbolize the golden

ratio

x + y

yx

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 aurea »In 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 :

Rectangles AEFD and BEFC are golden rectangles, the ratio of their

sides is equal to

THE GOLDEN SPIRALSo, we can build series of golden rectangles and therefore plot a golden spiral. That spiral can be found in some shells.

THE GOLDEN TRIANGLESThey are isosceles triangles, such as the ratio between two sides is

equal to . There are two types:

THE REGULAR PENTAGONEach angle is equal to 108°, it is made up of golden triangles

and, if the sides length is equal to 1, its diagonal is equal to

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.

A CONTINUED FRACTION

The most irrational amongst irrational numbers ….

THE GOLDEN RATIO IN ARTMany people looked for the golden ratio into art works

Is the Parthenon design really based upon the golden ratio ? Drawings recommended by those who sustain that theory seem complicated …

THE GOLDEN RATIO IN ARTSeurat’s painting

Based upon the golden ratio?

Or by dividing in two, then in two, the in two …. i.e. in

THE 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 «  cité Radieuse » in

Marseille

THE 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 point

Could You find a golden point here?

« The crab with the golden claws ».

Hergé

FIBONNACCI SEQUENCELeonardo Fibonnacci (12e/13e century) Italian Mathematician, also named Leonardo PisanoHe has worked on the following sequence of natural numbers• The first two equal 1• A 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 ….

FIBONNACCI’S 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 rabbits• A couple of adults doesn’t die, and gives birth to a couple of babies

THE GOLDEN RATIO IN NATURE

Geospace modelling

• The golden angle divides a circle into two arcs whose ratio is .

As the most « irrational number », it’s 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

Fibonnacci’s sequence.

GOLDEN RATIO AT THE CATHEDRALS TIME

You can find again the terms of the Fibonnacci’s

sequence

Cathedrals 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 cmCoudée 233 lignes 52,36 cm

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

G

T

N

 

 

 

 

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