• Fibonacci series is an Integer sequence.
first two numbers are - 1 and 1, or 0 and 1 (depending on
• closely related to Lucas numbers and Golden
• 1st in Indian mathematics.
• Invented by Leonardo Fibonacci.
• Discovered in an investigation of Reproduction of
Reproduction of rabbits
•Integer sequence is a sequence or an ordered list of
For example, a sequence 0, 1, 1, 2, 3, 5, 8, 13, …
(the Fibonacci sequence)
is formed by starting with 0 and 1 and then adding any two
consecutive terms to obtain the next one. The sequence 0, 3, 8,
15, … is formed according to the formula n2 − 1 for the nth
•Each Lucas number is defined to be the sum of its two
immediate previous terms, thereby forming a Fibonacci
integer sequence. The first two Lucas numbers are L0 = 2 and L1
= 1 as opposed to the first two Fibonacci numbers F0 = 0 and F1
= 1. Though closely related in definition, Lucas and Fibonacci
numbers exhibit distinct properties.
•The ratio between two consecutive Lucas numbers converge to
the golden ratio.
Two quantities are in the golden ratio if their ratio is the same as
the ratio of their sum to the larger of the two quantities
• Leonardo Fibonacci (1170-1240),lived in
• He was a Catholic Christian in religion.
• Most talented western mathematician
of middle age.
• Introduced the Hindu-Arabic
• Introduced Fibonacci Series in his book
Liber Abaci (1202).
• Introduced Digital Notation to Europe.
Fibonacci In Mathematics
• Integer sequence.
• Fibonacci spiral: an approximation of the
golden spiral created by drawing circular arcs
connecting the opposite corners of squares in the
• A Recurrence relation (an equation that
recursively defines a sequence.)
• Fibonacci numbers occur in the sums of "shallow"
diagonals in Pascal's triangle.
• Fibonacci number is the length of the hypo-tenuse of
a right triangle with integer sides. Ex: 1,2,3,5
Fibonacci In Nature
• Appears in Biological settings
• such as the fruit sprouts of a pineapple, the, an uncurling fern and the arrangement of a
Arrangement of leaves on a stemFibonacci in Galaxies
Fibonacci In Art
Birth of Venus
67.9 in × 109.6 in
• Intimately connected with golden ratio.
• If we start the series with 0, we get golden ratio. If we start from 1,
we get golden spiral.
• Used in art. Such as – Birth Of Venus by botticelli, Parthenon etc.
• The ratio 1:1.618 is very pleasing to eyes.