Caricatures improved

I was practicing on caricatures these days. I was not comfortable with the 3d look in them and followed Andrew Loomis method very carefully. Thanks to Andrew Loomis for his wonderful book "Drawing head and hands". The book is excellent and it took a while to pick up his concepts.
Now I am a bit more confident.
So, here is some caricatures I drew today:-

Caricatures

I always drew caricatures with face facing straight. Then today a thought came up. I was always confident in side views. But straight I was not. Then I thought mix these two and draw 3/4 views.
The above caricatures I made just now. From memory. The above have simple lines. But I plan to extend them to highly complex 3/4 views. In coming months.
So I have frozen my art desires. Caricatures 3/4. Cartoons.
Best wishes to all of my blog readers.

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.