Golden ratio

the golden ratio

1.618...

start

What is the golden number ?

- Since :

- the golden ratio : studied by the Greeks, Euclid in III millenary BC.

- We have :

- Multiplying by Φ gives :

- Let's write Φ a greek letter for that ratio :

- Using the quadratic formula to solve this equation we get a positif root :

- The left hand side of the equation can be written as :

Many properties

- It's easy to show that :

- Let's find a number A such that :

A =

= 1 + A

A 2=

1 +

So, A 2= A + 1 is

and the solution is Φ !

the rectangular version of the golden ratio

the golden spiral !

. - THE GOLDEN RECTANGLE -

the FIBONACCI SEQUENCE

A man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits are produced from that pair in a year, if it supposed that every month each pair produces a new pair, which from the second month onwards becomes productive?"

Leonardo Da Pisa, aka Fibonacci ( 1 170 - 1240 )

We get : 1, 1, 2, 3, 5, 8, 13, 21, 34...

the FIBONACCI SEQUENCE and the golden number

What is the connection with the golden number ?

--> by succesively dividing each term in the Fibonacci sequence by the previous term we come closer to the golden ratio !

the GOLDEN RATIO IN Architecture

the GOLDEN RATIO IN ARTS

Leonerdo Da Vinci and the Vitruvian man

The Parthnon in Athens : Construction : - 447, most important buildnig of Classical Greece.

The great Pyramid of Giza in Egypt

The ratio of the square to the radius of the cercle corresponds to the golden ratio.

Le Corbusier and the "cité radieuse" in France :

the GOLDEN RATIO IN Architecture

the GOLDEN RATIO IN Architecture

72 m height, in the Indian city of Algra, built in 1632.

the GOLDEN RATIO IN NATURE

The Fibonacci sequence appears in leaves, branches, pinapple skin, broccoli....

The Fibonacci sequence and the golden number are everywhere !

thank you !

it was a presentation by the twins amel and sofia !