Welcome to a presentation
on the Fibonacci Sequence.
The Fibonacci sequence is the number list shown here,
though some sources don't include the zero.
To create the Fibonacci sequence, we start with 0 and 1,
and then each term is the sum of the two previous terms.
Starting with 0 and 1,
the next term is 1 because 0 plus 1 is 1.
The next term is 2 because 1 plus 1 is 2.
The next term is 3 because 1 plus 2 is 3.
The next term is 5 because 2 plus 3 is 5, and so on.
We can say that the first term is a sub zero,
the second term is a sub one,
the third term is a sub two and so on.
Using this notation, we can make a recursive formula
for the Fibonacci sequence,
where a sub 0 equals 1, a sub 1 equals 1,
and therefore a sub n equals a sub n minus 2
plus a sub n minus 1.
For example, a sub three, because n is equal to 3,
and 3 minus 2 is 1, and 3 minus 1 is 2,
a sub 3 equals a sub 1 plus a sub 2,
which in this case, is 1 plus 1, which equals 2,
which is the fourth term in the Fibonacci sequence.
The Fibonacci sequence is probably
the most famous number sequence.
It is named after the Italian mathematician
Leonardo Pisano of Pisa, known as Fibonacci.
His 2002 book Liber Abaci introduced the sequence
to Western European mathematics,
although the sequence had been discovered
earlier in Indian mathematics.
In Fibonacci's book, the Fibonacci sequence emerged
as the solution to the following rabbit problem.
A newly-born pair of rabbits, 1 male, 1 female,
are put into a field.
Rabbits are able to mate at the age of 1 month,
which means at the end of the second month,
a female can produce another pair of rabbits.
Supposed the rabbits never die
and the female always produces 1 male and 1 female.
Determine how many pairs of rabbits after each month.
So looking at the diagram here, we have the initial pair
of rabbits that can reproduce after two months.
So after two months,
this pair of rabbits reproduce this pair of rabbits,
and notice now there are two pairs of rabbits.
But remember, this pair of rabbits
can only reproduce after two months,
which means for the next month,
this pair of rabbits reproduce again,
reproducing this pair of rabbits,
and now there are three pairs of rabbits.
And for the following month, this pair of rabbits
can now reproduce, producing this pair of rabbits,
and the original pair of rabbits can reproduce again,
producing this pair of rabbits.
And this pair of rabbits cannot yet reproduce.
Notice there are now five pairs of rabbits.
Continuing, the pairs of rabbits after each month
give us the Fibonacci sequence.
Notice how for this Fibonacci sequence,
zero is not included.
One of the reasons the Fibonacci sequence is so popular
is that the numbers appear all around us.
For example, the number of petals
in most flowers are Fibonacci numbers.
For example, here we have a flower with five petals,
five is a Fibonacci number.
Here we have a flower with eight petals,
eight is a Fibonacci number.
Here we have a flower with 21 petals,
again 21 is a Fibonacci number.
Also notice how, if we slice an apple horizontally,
we often see five points,
where five is also a Fibonacci number.
The Fibonacci sequence is also related to the golden ratio.
The golden ratio is the limit of the ratios
of successive terms of the Fibonacci sequence.
So the golden ratio is equal to phi or phi,
which is exactly equal to the quantity
1 plus the square root of 5 divided by 2.
This is discussed in another video,
but if we take the ratio of successive terms
of the Fibonacci sequence,
these ratios do approach the value of the golden ratio.
1 divided by 1 is equal to 1, 2 divided by 1 is equal to 2,
3 divided by 2 is equal to 1.5,
8 divided by 5 is approximately 1.67,
if we continue taking these ratios,
these values do approach the golden ratio,
which is approximately 1.618.
Notice how this ratio is approximately 1.619.
The Fibonacci sequence is also related to the golden spiral.
A Fibonacci spiral approximates the golden spiral,
using quarter-circle arcs inscribed in squares
of integer Fibonacci-number sides, shown for square sizes
1, 1, 2, 3, 5, 8, 13, and 21, shown here.
So this Fibonacci spiral approximates the golden spiral,
where the golden spire is a log spiral,
whose growth factor is phi or phi, the golden ratio.
This spiral gets wider by a factor of phi
every quarter turn.
So again, the Fibonacci spiral
approximates the golden spiral.
And we often see these spirals in nature.
Many plants grow in spirals.
Often the number of spirals is a Fibonacci number
and the spiral resembles the Fibonacci spiral.
Here we have a plant or an agave,
we can see the spirals outlined in red.
And notice how if we count the spirals,
we have one, two, three, four, five spirals,
and five is a Fibonacci number.
Here's the bottom of a pine cone,
and again we can see the spirals.
If we count the number of spirals, we have one, two, three,
four, five, six, seven, eight, nine,
10, 11, 12, 13, and 13 is also a Fibonacci number.
We can also see the spirals here in this flower
as well as this vegetable.
And here are some additional examples
of the Fibonacci spiral in nature.
Here we have the spiral of a shell, the spiral of a plant,
the spiral of a storm or a weather pattern,
and here we have a spiral in space.
There are many other interesting topics
related to the Fibonacci sequence,
so you may want to do some additional research.
Thank you for watching.