Wallis' product exploration
02-09-2020, 05:44 AM
Post: #8
 pinkman Senior Member Posts: 432 Joined: Mar 2018
RE: Wallis' product exploration
(02-08-2020 08:58 PM)Allen Wrote:
(02-08-2020 07:53 PM)Albert Chan Wrote:  That is because the formula had this denominator (squared!)
$$\Large 2\prod _{k=1} ^{k=n} {4k^2 \over 4k^2-1} = {2^{4n+2} \over 4n+2} รท \binom{2n}{n}^2$$

Interesting!!

Since the central binomial coefficients are related to Catalan numbers, Wallis' product $$\pi$$ approximation for each $$n$$ is inversely proportional to the number of ways a regular $$n$$-gon can be divided into $$n-2$$ triangles.

As $$n \rightarrow \infty$$ then the $$n$$-gon shape approaches a circle.

Wait, $$\pi$$ is related to circles because of triangles?

Really interesting. And funny
So, this apparently abstract formula of Wallis is the same approach to find PI than any other method: converge to a circle. Well, what did I expect? No PI without circles!
It also leaded me to wake up my memory about binomial coefficients, that I had a bit (completely) forgotten...

Thibault - not collector but in love with the few HP models I own - Also musician : http://walruspark.co
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 Messages In This Thread Wallis' product exploration - pinkman - 02-07-2020, 10:55 PM RE: Wallis' product exploration - Albert Chan - 02-08-2020, 02:13 AM RE: Wallis' product exploration - pinkman - 02-08-2020, 02:42 PM RE: Wallis' product exploration - Allen - 02-08-2020, 07:13 PM RE: Wallis' product exploration - Albert Chan - 02-08-2020, 07:53 PM RE: Wallis' product exploration - Allen - 02-08-2020, 08:58 PM RE: Wallis' product exploration - pinkman - 02-09-2020 05:44 AM RE: Wallis' product exploration - Allen - 02-08-2020, 08:13 PM RE: Wallis' product exploration - Allen - 02-09-2020, 01:08 PM RE: Wallis' product exploration - Allen - 02-09-2020, 01:57 PM RE: Wallis' product exploration - Allen - 02-09-2020, 03:08 PM RE: Wallis' product exploration - pinkman - 02-09-2020, 02:14 PM RE: Wallis' product exploration - Gerson W. Barbosa - 02-09-2020, 10:58 PM RE: Wallis' product exploration - Gerson W. Barbosa - 02-11-2020, 12:53 AM RE: Wallis' product exploration - EdS2 - 02-10-2020, 10:35 AM RE: Wallis' product exploration - pinkman - 02-11-2020, 10:02 AM RE: Wallis' product exploration - Gerson W. Barbosa - 02-11-2020, 04:11 PM RE: Wallis' product exploration - Gerson W. Barbosa - 02-11-2020, 10:48 PM RE: Wallis' product exploration - pinkman - 02-12-2020, 10:01 PM RE: Wallis' product exploration - Gerson W. Barbosa - 02-13-2020, 12:17 AM

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