Introduction to Fibonacci number
In mathematics, the Fibonacci numbers are the numbers in the
following integer sequence:
A tiling with squares whose sides are
successive Fibonacci numbers in length
By definition, the first two numbers in the Fibonacci
sequence are 0 and 1, and each subsequent
number is the sum of the previous two.
In mathematical terms, the sequence Fn of Fibonacci
numbers is defined by the recurrence relation:
with seed values,
A Fibonacci spiral created by drawing circular arcs connecting the opposite corners
of squares in the Fibonacci tiling; this one uses squares of sizes 1, 1, 2, 3, 5, 8, 13, 21,
and 34. See golden spiral.
The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci.
Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics,
although the sequence had been described earlier in Indian Mathematics.
(By modern convention, the sequence begins with F0 = 0. The Liber Abaci began the sequence with F1 = 1,
omitting the initial 0, and the sequence is still written this way by some.)
Occurrences in mathematics
The Fibonacci numbers are the sums of the "shallow" diagonals
(shown in red) of Pascal’s Triangle
The Fibonacci numbers can be found in different ways in the sequence
of binary strings.
•The number of binary strings of length n without consecutive 1s is the
Fibonacci number Fn+2. For example, out of the 16 binary strings of
length 4, there are F6 = 8 without consecutive 1s – they are 0000,
0100, 0010, 0001, 0101, 1000, 1010 and 1001. By symmetry, the
number of strings of length n without consecutive 0s is also Fn+2.
•The number of binary strings of length n without an odd number of
consecutive 1s is the Fibonacci number Fn+1. For example, out of the
16 binary strings of length 4, there are F5 = 5 without an odd number of
consecutive 1s – they are 0000, 0011, 0110, 1100, 1111.
•The number of binary strings of length n without an even number of
consecutive 0s or 1s is 2Fn. For example, out of the 16 binary strings of
length 4, there are 2F4 = 6 without an even number of consecutive 0s
or 1s – they are 0001, 1000, 1110, 0111, 0101, 1010.
Probably most of us have never taken the time to examine very carefully
the number or arrangement of petals on a flower. If we were to do so,
several things would become apparent. First, we would find that the
number of petals on a flower is often one of the Fibonacci numbers. Onepetalled ...
white calla
and two-petalled flowers are not common.
euphor
b
Three petals are more common.
Trillium
There are hundreds of species, both wild and
cultivated, with five petals.
columbin
e
Eight-petalled flowers are not so common as fivepetalled, but there are quite a number of wellknown species with eight.
bloodroot
Thirteen, ...
black-eyed susan
Twenty-one and thirty-four petals are also quite
common. The outer ring of ray florets in the daisy
family illustrate the
Fibonacci sequence
extremely well. Daisies with 13, 21, 34, 55 or 89
petals are quite common.
Ordinary field daisies have 34 petals ... a fact to be
taken in consideration when playing "she loves
me, she loves me not". In saying that daisies have
34 petals, one is generalizing about the species but any individual member of the species may
deviate from this general pattern. There is more
likelihood of a possible under development than
over-development, so that 33 is more common
than 35.
The association of Fibonacci numbers and plants
is not restricted to numbers of petals. Here we
have a schematic diagram of a simple plant,
the sneezewort. New shoots commonly grow out
at an axil, a point where a leaf springs from the
main stem of a plant.
shasta daisy with 21
petals
If we draw horizontal lines through the axils, we
can detect obvious stages of development in the
plant. The main stem produces branch shoots at
the beginning of each stage. Branch shoots rest
during their first two stages, then produce new
branch shoots at the beginning of each
subsequent stage. The same law applies to all
branches.
Since this pattern of development mirrors the
growth of the rabbits in Fibonacci's classic
problem, it is not surprising then that the number
of branches at any stage of development is a
Fibonacci number.
Furthermore, the number of leaves in any stage will also be a
Fibonacci number.
Our hand shows the Fibonacci Series
Each section of your index finger, from the tip to the base of the wrist, is larger than the
preceding one by about the Fibonacci ratio of 1.618, also fitting the Fibonacci numbers 2, 3, 5
and 8.
By this scale, your fingernail is 1 unit in length.
Curiously enough, you also have 2 hands, each with 5 digits, and your 8 fingers are each
comprised of 3 sections. All Fibonacci numbers!
The ratio of the forearm to hand is Phi
Your hand creates a golden section in relation to your arm, as the ratio of your forearm to your
hand is also 1.618, the Divine Proportion.
Even your feet show phi
The foot has several proportions based on phi lines, including
:
1) The middle of the arch of the foot
2) The widest part of the foot
3) The base of the toe line and big toe
4) The top of the toe line and base of
the "index" toe
Prepared by:
Shubham & Ankit
Class :- +1
Drv d.a.v. Centenary public school ,,
phillaur (india).