In school when we are taught math
 
it is usually in the form of theories , formulae
and functions
Very rarely we meet such teachers
who tell us the importance of mathematics in life
Infact there is a deep relation between math and nature.
 
for instance
cicada insect has more than 200 species
 
and almost  100 species can be found in India.
this insect lives underground
and comes out every few years
lets say in "N" years.
this "N" usually are primes numbers.
Honeycomb's
shape is usually hexagonal
with six sides
which is a very efficient way
to store honey
Corona virus pandemic which is in news these days
also follows a logistic curve
And the most amazing one is the
Golden Ratio
which is somewhere equal to 1.618
this ratio can be found around us
and everywhere
for example: in the world of honey bees
male and female population of bees
also follows Golden ratio.
And in humans facial beauty measurement
also uses Golden ratio.
 
Infact this Golden ratio
can be seen in our DNA
So now you will ask, where did this ratio come from?
the answer is : Fibonacci Series
In this series
as you go ahead
every two consecutive number's ratio
will start reaching toward Golden ratio.
So lets move forward
and understand how this series is formed.
1,2,3,5,8
13, 21,34, 55...
are called Fibonacci series.
These numbers
depict a pattern
Every number is a result of summation of previous two numbers
 
For instance, if we start the series
from 1, then first two numbers
will be 1 & 2
adding them gives us the third number, 3
Similarly adding 2 & 3 will give
the fourth number, 5
adding 3 & 5 gives fifth number , 8
and in this way the series continues.
 
These numbers are globally known as Fibonacci series.
 
But very few people know this,
around 100 yrs before Fibonacci,
Hemchand spoke about it in India
And 400 yrs before that
an Indian Poet Virhanka
around 700 AD
proposed this concept.
So based on chronology
we will call this series
Virhanka numbers.
So now we will try to understand
how this series came into existence.
Virhanka was Sanskrit Poet
and a mathematician.
And he wanted to understand that
if the length of a poem is defined
then in how many ways can it be written?
Usually in Sanskrit shloka
two types of sounds are used.
First one is called : Laghu
second : Guru
Laghu takes less time to speak
and Guru takes more time in camparision to laghu
 
For example lets assume
Dum is a Guru sound
and Da is Laghu sound
now in sanskrit
sounds are measured in the form of Matras
so Dum
being a Guru sound will denote 2 matras
and Da being a Laghu sound will be 1 matra.
 
So lets understand Virahanka's question once again
 
What he meant was that
for a given length of poem
if the number of Matras and their types are defined
then in how many ways can it be written?
So lets assume,
this poem is of 4 matras
and we have already established that Dum is 2 matras
and Da is 1 matra
 
So lets see in how many ways can we write them
 
For this case of 4 matras in a poem
we can simply use 2 DUM
DUM  DUM
Or we can replace one DUM
with 2 Da
since Da is just one matra
So now it becomes DUM  DA  DA
and now this can
be rearranged in different styles
 
DA  DUM  DA
DA  DA  DUM
Now we replace the other DUM
with 2 more DA
DA  DA  DA  DA
 
These can also be rearranged but the result will be same.
 
DA  DA  DA  DA
 
SO now if we see them together
those 4 matras
will result in how many ways?
DUM DUM
DUM DA DA
Da DUM DA
DA DA DUM
DA DA DA DA
in total 5 ways.
So now if we consider the number of matras as "n"
 
then the function will be Vn
So the above example can be written as
V(4) = 5
So lets start apply this from the beginning
If we have 1 matra
so in how many ways can we write it?
Only 1, which is 1 laghu.
because for Guru we need atleast 2 matras
DA
V(1) = 1
Now if we have 2 matras
then how many ways? Either 2 laghus
or 1 guru
DA DA or just DUM
V(2) = 2
 
Now if we have 3 matras
then its is DA DA DA
DA DUM
DUM DA
V(3) = 3
Now if we have 4 matras ?
This, we have solved earlier as well
V(4) = 5
Are you now able to see the
pattern in Virahanka Numbers?
V(1) = 1
V(2) = 2
V(3) = V(2) + V(1)
Which is
2+1 = 3
similarly
V(4) = V(3) + V(2)
3+2 = 5
 
Now we can use the above formula
and calculate for V(5), V(6), V(7),V(8)
 
 
and then can conclude the above pattern
in the shape of a formula.
 
Vn=Vn-1 + Vn-2
 
 
Now lets prove the authenticity of this formula.
 
For proof Virahanka
used a small shlok
which meant
"That each shlok will either end with a Guru or a Laghu"
"That each shlok will either end with a Guru or a Laghu"
thats it.
He knew he was right.
SO this basically means
that if we have "n" matras with us
then at the very end of every shlok
there will be either Laghu or Guru
So now lets take all the shloks
which end with a Laghu
and remove that last laghu from there.
And we already know that Laghu means 1
so now we are left with how many matras in that shlok?
n-1
and
in how many ways can we write it?
Vn-1
Similarly
we remove the Guru from the shloks which end with a Guru
So we will be left with n-2 matras
sinve Guru is 2
and in how many ways can it be written?
Vn-2
Lets take another example
V(4)
in this example I will use colors.
so that its easier to understand
so lets take V(3) as green
V(2) blue
and the new word which will be added for V(4)
there we will use yellow
Now we already know that
V(4) = V(3) + V2)
and V(3) is  DA DA DA
DUM DA   &   DA DUM
now lets see
those type of V(4) which end with Laghu
that would be
V(3) + DA
as you can see here
in green there is V(3)
the newly added ones are in Yellow
Now we look into V(2)
DA DA
and DUM
Now similarly lets see
those type of V(4) which end with DUM
 
and this makes
V(2) + DUM
and you can see here
V(2) is in Blue
and in yellow we have newly added DUM
so in this ways V(4) can be created
and is equal to 5
which we proved earlier
hence,
Vn = Vn-1 + Vn-2
can be used
and you can create and check the whole Virahanka numbers.
It is surprising to see that everywhere
around us there are Virahanka numbers.
Even Pine cone
Have you ever counted their spirals?
they are usually Virahanka numbers
Daisy flowers
petals of daisy are also Virahanka numbers
Is it not amazing?
All we have to do is look around
nature and math would become so easy to understand
As today you have learnt that
this whole Virhanka numbers was discovered
as an answer to a question
similarly many theories in maths are inspired by life around us,
Next time if someone asks you about maths and its relation with our life
show them this video .
We hope this was helpful
and you can use this in future.
So before you leave we have 2 questions for you
1st
If the shlok has to use 3 types of sound
Laghu 1, Guru 2 & Maha 3
so then
what would be the formula for Vn?
Vn = ???
2nd ques
In the 1st ques  if there are 8 matras then what would the answer?
 
V8 = ????
Do share the answer with us in comment section.
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Thank You
