About some pi expressions

Most of the formulas for pi are based on mathematical functions. But a few of them are not. These are driven by intuitions and are based on numbers, prime numbers or square roots.
Here are some of them :-

Two of them from Euler. And one from Viete.
Yesterday evening I was thinking hard on root 2, root 3, root 4 . . . . . . etc.
Then I found that

The 12th term was 0.25908. The 13th term was 0.52634. Their average is 0.39271. And 0.39271 is about pi/8.
The results in Wolfram from root(2) to root(5000) and root(5001) are:-

The average sway is (0.40217207+0.38803135)/2 = 0.3951017. It is pi/7.95135.

So the sway remains at about pi by 8.

Indeed the Gauss Error Function observed here.

Rediscovering 1,3,6,10 . . .

1,3,6,10,15 . . . already exists. 1,3,6,10,15 . . . is the triangle series.
I didn't know that. I rediscovered it. I applied rational growth and made some formulas. So the series was rediscovered.
I found another thing yesterday. The square series 1,4,9,16,25 . . . relationship with 1,3,6,10,15 . . .
Subtracting term by term each of the numbers of square series with the triangle series creates a series of terms 0, another triangle series.
1 - 1 = 0
4 - 3 = 1
9 - 6 = 3
16 - 10 = 6
25 - 15 = 10.
and so on . . .

Mathematical Philosophy continues

Before one takes any step one should outline steps of thinking. The deepest steps of my thoughts are:
1. There is zero. Vortex. Gravity or human driven.
2. There is basic c. Complex number center.
3. There is z. Human complex interpretation.
4. There is area. The beginning.
And/Or
1. There is zero. Vortex. Gravity or human driven.
2. There is basic c. Complex number center.
3. There is z. Human complex interpretation.
4. A division. 2 and 2. Or 1 and 1. Bi-directional.
And/Or
1. There is zero. Vortex. Gravity or human driven.
2. There is basic c. Complex number center.
3. There is z. Human complex interpretation.
4. A delta triangle. 3 and 1. 3 are sides. 1 is direction.
All the three are important. And are sort of recursive.

My truth is of and for Indians

I wrote about Laurent Series and the completion of a round in a sports track and beginning of 1.
I feel Abel transform is similar to Laurent series in some way.
Also, Harishchandra (Indian mathematician) transform is similar but in parabolic form.
My point is this. I am talking of an India which I see.
Only if you have food can one complete a round around an olympic track.
My # expressions is not about completion. It may be imagined as height by length. And growth there in.
So according to my model if the villagers of Kalahandi (where people died due to famine) had a good relationship with Tanjoreans (rice growing area in Tamil Nadu) which is a rational number; and this rational number had grown then the death at Kalahandi could have not happened.
I am quite serious when I am writing this.
Please support me.

The beginning - Laurent series

Carl wants to count the number of rounds of the olympic track he is able to complete by running.
Tracy is observing and counting.
Only when Carl encloses an area does Tracy begins the count with 1 round completed.
So there is an area. There is n. There is n+1(next round).  There is periodicity(sigma series). Then there is the beauty in Laurent expression. 

Now What?

The mathematics I was writing about is over.
But mathematics is a big field. There are areas like discrete mathematics, algebra, calculus in detail, set theory,  hyperbolic geometry, algorithms etc. So a lot of scope exists for everyone.
I would be reading more from "Theory of Complex Functions" book.

Pentagon and heptadecagon

A side of a polygon is 2a× sin(180/n) where n is the number of sides of a polygon.
When I drew a pentagon I used the formula
Root(2) - (3/4)(Root(3) - Root(2)). This requires only a compass. 1.175 is the side length of a pentagon if radius is 1.
Then I read in Wikipedia that as a kid Gauss drew a heptadecagon (17 sided polygon) using a compass. The full name of Gauss is Johann Carl Friedrich Gauss. Gauss was a child prodigy and made contributions due to which he is considered "the greatest mathematician since antiquity". He was also called "the foremost of all mathematicians". The erstwhile cgs unit of magnetic flux density was named after him. Gauss. Many web sites over internet say Gauss is the "Prince of Mathematics". And the "King of Mathematics" is Leonhard Euler.
Well I tried the adaptation of heptadecagon with a formula similar to pentagon.
(Root(14) - (3/4)(Root(15) - Root(14)))/10. Dividing by 10 needed here. (This cannot be obtained with compass. Dividing by 10) In a pentagon 5 you had 2 & 3. For heptadecagon 17 you have 14 and 15.  The value is 0.364 while actually the value is 0.367. The side length of heptadecagon if radius is 1.
So a similar formula for both pentagon and heptadecagon exists. However dividing into 10 is not possible with a compass only. One needs setsquares or drafter. Anyway this method of mine needs a very large paper.

About Ramanujan

Ramanujan was a great mathematicians. I have no doubts about it. He made what are now known as Ramanujan theta functions, Ramanujan tau functions and Ramanujan Sato series.
I have applied myself and have explained these three important functions in my blogs.
Ramanujan had made great contribution in partition of numbers. Eg. 4 can be expressed as 4,3+1,2+2,2+1+1,1+1+1+1.
Now, my series of 1,3,6,10,15 . . . is a special case of partition. 1,1+2,1+2+3,1+2+3+4,1+2+3+4+5, . . . .
I have demonstrated the importance of 1,3,6,10 . . . in my blogs.
So partition of numbers as an Arithmetic Progression is important.
I have demonstrated the growth of rational numbers too.
Partition of numbers in Arithmetic Progression and rational number growth have a connection.
I am interested in further studies and/or work.

The Kashmir solution

Let us understand history and early British plans.
Sir Radcliffe and Lord Mountbatten partioned India. Accession of individual kingdoms and provinces to  India was essential. Most of India was made that way. The king of Kashmir accepted India. Then there were important foreign presence provinces and structures like places in Goa, Pondicherry, Chandigarh, Delhi which were made union territories.
Now since Pakistani flag and Pakistani presence is high in Kashmir; the Government of India has rightly made union territories in the state of Jammu and Kashmir as Pakistan is a foreign country.

Looking back and ahead

I am back in Ranchi.
It was in November 2010, that I made the blog Measuring Behaviour. During those times I managed to express pi approximately as 22/7. I also had expressions on the mathematical constant e. Frankly there were two things that I wondered. 2n and n. World is unfair(n). World is fair(n). And you measured the behaviour(2n). Gradually I started expressing better. I made a few infinite series expressions. And currently I am with the book "Theory of Complex Functions". Hoping to learn the methods and lemmas.
Later, I realised you have to consider unfair(n) and fair(n) together for measuring behaviour(2n). Expressing in terms of infinite expression, I got

Later I fine tuned and got better pi expressions. These are

I later realised that comb(2n,n) is great in expressions of irrational numbers too. I got

Since I got root 5, I expressed the most famous golden ratio. This what I got

Now. As I wrote above, unfair(n), fair(n), measuring behaviour(2n) is important. But one cannot deny that one looks at the world with only one thing in mind. Either fair. Or unfair. Is it not?
So I tried n and 2n. This is what I got

This has opened a new world to me. For one when dealing with n and 2n and not n,n and 2n, you get an expression which has both pi and e terms. In one expression you have both pi and e in the numerator and in the other(alternated negatives) you get pi in the numerator and e in the denominator.
This gives a lot of insight into the behaviour of a man. If he looks at the world as unfair or fair (one of these) he gets an equation which is absolutely different from if he looks at the world as both fair and unfair, both together.
I am also pasting the definition of error function (of Carl Friedrich Gauss) and that of imaginary error function too.