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THE FIBONACCI SEQUENCE
AND
THE GOLDEN RATIO
Created by: Szabó Miléna Dóra

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FIBONACCI (Leonardo Pisano)
• Fibonacci was an Italian
mathematician considered
to be „the most talented
Western mathematician of
the Middle Ages”.
• The name he is commonly
called, "Fibonacci" is short
for "figlio di Bonacci"
("son of Bonacci")

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THE FIBONACCI SEQUENCE
The Fibonacci Sequence is the series of
numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34,55,89,144 ...
The rule:
The next number is found by adding up the
two numbers before it. (n≥2)
xn = xn-1 + xn-2

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EXPLICIT FORMULA
Explicit formula means, a formula that allows
direct computation of any term for a sequence.
Fibonacci sequence’s explicit formula:

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WHERE DOES THE SEQUENCE
COME FROM?
Fibonacci considers the growth of an idealized
(biologically unrealistic) rabbit population,
assuming that:
• a single newly born pair of rabbits (one male, one
female) are put in a field;
• rabbits are able to mate at the age of one month
so that at the end of it’s second month a female
can produce another pair of rabbits;
• rabbits never die and a mating pair always
produces one new pair (one male, one female)
every month from the second month on.

6. THE RABBITS
Month 1st 2nd 3rd 4th 5th …th
pair 1 1 2 3 5 …

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FIBONACCI SPIRAL
Click here!
It’s moving

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IN NATURE
Patterns created by the fibonacci sequence can
be found throughout nature.
In petal patterns:

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IN THE SUNFLOWER:
IN FRUITS: IN PLANTS:
Solar cell inspired by
the sunflower:

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THE GOLDEN RATIO
We find the golden ratio
when we divide a line into
two parts so that: the
whole length divided by
the „a” part is also equal
to the „a” part divided by
the „b” part.

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π φ e
Phi φ ≈ 1.6180339887498948420 …
This irrational number is the Golden ratio. Similarly as the
well known π and e.
There is a special relationship between the Golden Ratio and
the Fibonacci Sequence.
When we take any two successive (one after the other) Fibonacci
Numbers, their ratio is very close to the Golden Ratio. In fact,
the bigger the pair of Fibonacci Numbers, the closer the
approximation.

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PENTAGRAM
The golden ratio hidden inside the pentagram.
a/b = 1,618...
b/c = 1,618...
c/d = 1,618...

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IN NATURE
The golden ratio is also found in nature.
• IN SHELLS:

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IN GALAXIES:

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IN HUMAN BODY
Leonardo Da Vinci:
Vitruvius de architectura

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IN ART
Just as the Golden Section is found in the design
and beauty of nature, it can also be used to
achieve beauty, balance and harmony in art and
design.
Hokusai:
The great wave off Kangawa
Edward Burne
Jones:
The golden
stairsLeonardo Da Vinci:
Mona Lisa

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IN ARCHITECTURE
Notre DameTaj Mahal
Parthenon

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The End
Click on the rabbits!