FIBONACCI

&

THE GOLD NUMBER

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)

He was one of the first people to introduce the Hindu-Arabic numbersystem 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 thirtysixFor example: two thousand and thirtysix

What did Fibonacci?...

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

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...)

1

3

4

67

2

5

Please!, choose the most aesthetic rectangle between the seven onesbelow…

But...why are so special these numbers?...

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 usingthis Golden Ratio in their works, believing this proportion to be aestheticallypleasing.

...6180,1b

a

But...why are so special these numbers?...

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…

But...why are so special these numbers?...

1,6180… The Golden Ratio! (can you believe it?)

The Fibonacci numbersand

The Golden Ratio

Mathemathics

Science

Architecture

Painting

Music Nature

Astronomy Sculpture

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

Achillea ptarmica (“sneezewort”)

Nature Petals on flowers

On many plants, the number of petals is a Fibonacci number:

white calla lily1 petal

Euphorbia2 petals

Trillium3 petals

Columbine5 petals

Bloodroot8 petals

black-eyed susan13 petals

shasta daisy21 petals

field daisies34 petals

Nature Petals on flowers

Fuchsia

4 petals… it isn’t a Fibonacci number!

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…

Nature Spirals in the Nature

Sunflower seeds Hurricane Galaxy

Sea shells

Nature Human body

Human ear: Fibonacci spiral

Human arm: Golden ratio

Human phalanx: Fibonacci numbers

Nature Human body

You can find many Golden Ratios in the human body

φ =

Science DNA doble helix

a

b

...6180,1b

a

Architecture Buildings & towers

Eiffel tower: Golden ratio

the Parthenon, in the Acropolis in Athens

Arts Painting

Three examples of Gold Ratio:

Man of Vitruvio

The Mona Lisa

Birth of Venus

Cards Credit cards

a

b

Cards Identity card