The Beauty of Pi; the Adolescence of Tau; and the Occasional Disharmony Between the Two

I happened across an article today - it was the 'Theorem of the Day' I saw in a Google+ post. It was about Machin's formula; which, from every other source I have ever seen, is usually expressed as:

However, on this particular website, it was expressed as:

It wasn't just this that I found slightly bizarre (by the way the bizarreness is not in the difference in how each presents arctan), but throughout the article they kept then referring to different ways of estimating Pi - ways that had been used in the history of mathematics - but calling it Tau/2. This is just so strange!!! I then noticed the "Tau Manifesto Compliant" logo in the bottom corner, but I just don't get why anyone would go this far!

I mean, I'm a fan of Greek letters - to the point where I've learnt a bit of koine Greek and so on - and I do like Tau. I think when you're using "2Pi" a lot, it is possibly easier (especially for coding) to have "Tau=2Pi" at the top and use tau from there in. But not at the expense of Pi! Not to keep using "Tau/2" - it's nonsensical!

I think what is lost here is the beauty of what Pi is - and has been - throughout mathematical history. It is not that we want to sanctimoniously hold on to Pi as if it has some intrinsic value - but rather that it is a beautiful constant which epitomises perfectly the direct harmony between order (the constant and necessary nature of Pi) in the natural world and chaos (the random and never repeating nature of the number itself) in the only way we can interpret it.

Which all reminded me of an ABSOLUTELY beautiful expansion - the Taylor series:

And how this can be manipulated to give:

Which is just beautiful.
Not only does it give us a function and way of estimating Pi (although for this purpose alone it is not the best one and there are many others which converge MUCH quicker than this one does!!!), it also uses the odd numbers as the denominators (and powers, though with 1 this is irrelevant) - which if you've read my last post, you'll know I like! ;)

Pi to 10,000 decimal places by Christian Ilies Vasile. Imagine a

string going from one digit on to the next: 3, 1, 4, 1, 5, 9, 2... and

changing colour each time it hit a different digit!!!

Nowadays, with our calculators that use Pi accurately and extensively, and computers which can generate Pi to any necessary number of digits, there is little need for exploration into this kind of systematic beauty that pi has always been a source of in mathematics.

The fact alone that the circumference of any circle will be Pi multiplied the diameter is astoundingly explanatory of what a circle is - a shape of which you could never reach the full extent of defining exhaustively. Just as the digits of Pi go on forever with no end, so do the intricacies of the natural world and even the humble circle.