My pi results

When one says pi by 4, immediately the Madhava of Sangamagrama formula crops up in mind.

When one talks of 1 by pi, the formulas of Ramanujan are thought of.

And pi by 3 into root 3? You must think of me. Kirtivasan.  😆😆😆🙏

Here are three instances of pi by 3 into root 3.

I want to develop deep mathematical reasons for why they are all equal. But unable to do so. And I have a lifetime ahead to read books and try and develop reasons for equivalence. It is very difficult for me to say why the three expressions are same.

However I would like to make a guess on intuitions. Remember my work on lifting okras with left hand. They were of gamma functions (n-1)!. So one expression is the gamma function activity of the hand. The other two are the fingers. Three consecutive and four consecutive fingers. I feel the golden ratio "speaks" of gamma function through pi constant. And it is of hand.

Have a very nice day dear buddies.  Be in great mood. Good things will happen. If nothing we shall together make great things.  

Stay with me !!