Fibonacci, golden number, free, Mathematics, Science, Architecture, Painting, Music, Nature, Astronomy, Sculpture

1. FIBONACCI
&
THE GOLD NUMBER

2. Who was Fibonacci?...
“The greatest European mathematician of
the middle ages“ was born in Pisa, Italy, in
1170 and died in 1250
He was known like Leonardo de
Pisa, Leonardo Pisano or
Leonardo Bigollo, but he was
also called “Fibonacci”
(fillius of Bonacci, his father’s
nickname)

3. He was one of the first people to introduce the Hindu-Arabic number
system into Europe, the positional system we use today.
It’s based on the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 with its decimal point
and a symbol for zero (not used till now)
But the most transcendental thing why he was known is by:
The Fibonacci numbers
Roman numeral Positional system
2036MMXXXVI
For example: two thousand and thirtysix
What did Fibonacci?...

4. Which are these numbers?...
By definition, the first two Fibonacci numbers are 0 and 1
These numbers are a numeric serie made with a simple rule of formation:
Each remaining number is the sum of the previous two

5. By definition, the first two Fibonacci numbers are 0 and 1
Each remaining number is the sum of the previous two
And then, the 15 first terms are…
Which are these numbers?...
These numbers are a numeric serie made with a simple rule of formation:
(Of course, there are infinite terms...)

6. Please!, choose the most aesthetic rectangle between the seven ones
below…
But...why are so special these
numbers?...

7. a
b
This rectangle is made using a special ratio between its long and its wide:
The Golden Ratio also called φ (phy).
At least since the Renaissance, many artists and architects have been using
this Golden Ratio in their works, believing this proportion to be aesthetically
pleasing.
( )ϕ...6180,1=
b
a
But...why are so special these
numbers?...

8. If we divide each term by the number before it, we will find the
following numbers:
From now onwards, the ratio is nearly constant, and equals…
1/1 = 1 2/1 = 2 3/2 = 1,5 5/3 = 1,666... 8/5 = 1,6
13/8 = 1,625 21/13 = 1,6153...
55/34 = 1,6176… 89/55 = 1,6181… 144/89 = 1,6179…
233/144 = 1,61805… 377/233 = 1,61802…
55/34 = 1,6176…
But...why are so special these
numbers?...
1,6180… The Golden Ratio! (can you believe it?)

9. The Fibonacci numbers
and
The Golden Ratio
Mathemathics
Science
Architecture
Painting
MusicNature
Astronomy Sculpture

10. Nature The plant branching
One plant in particular shows the Fibonacci numbers in the number of
"growing points" that it has.
Suppose that when a plant puts out a new shoot, that shoot has to grow two
months before it is strong enough to support branching. If it branches every
month after that at the growing point, we get the picture shown here.
1
1
2
3
5
8
13

11. Nature Petals on flowers
On many plants, the number of petals is a Fibonacci number:
white calla lily
1 petal
Euphorbia
2 petals
Trillium
3 petals
Columbine
5 petals
Bloodroot
8 petals
black-eyed susan
13 petals
shasta daisy
21 petals
field daisies
34 petals

12. 1
1
2
3
5
8
13
Nature Spirals in the Nature
Add another square below this, with a size of 1 unit
Add another to the left with a size of 2 unit
Add another on top, with a size of 3 unit
Add another to the right, with a size of 5 unit
Repeat these operations with 8, 13, 21...
Draw a square, with a size of 1 unit
Then, draw an spiral, starting from the outer edge to the opposite…

13. Nature Spirals in the Nature
Sunflower seeds Hurricane Galaxy
Sea shells

14. Nature Human body
Human ear: Fibonacci spiral
Human arm: Golden ratio
Human phalanx: Fibonacci numbers

15. Nature Human body
You can find many Golden Ratios in the human body
φ =

16. Science DNA doble helix
a
b
...6180,1=
b
a

17. Architecture Buildings & towers
Eiffel tower: Golden ratio
the Parthenon, in the Acropolis in Athens

18. Arts Painting
Three examples of Gold Ratio:
Man of Vitruvio
The Mona Lisa
Birth of Venus

19. Cards Credit cards
a
b

20. Cards Identity card

21. Thank you