Where is the error on this approximation to $\pi$

I came across this website (see also) where the author (a supposedly alternative mathematician) claims to have a better approximation to $\pi$.

$$\pi\approx 3.1547…$$

Can someone tell me where is the error in his reasoning? (Sorry for the pessimistic viewpoint)

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Perhaps you can explain what he is doing.. I don't get it. – picakhu Sep 18 '12 at 12:30
@picakhu I see what you see. Sorry, I do not know any additional details. – pion Sep 18 '12 at 12:31
Just because someone is claiming to have done something great, does not mean he is wrong. If you look closely at what he is trying to say, you will find his reasoning make sense. – user43704 Oct 5 '12 at 10:08
If you look closely at any carefully crafted argument, it will usually make sense. Making sense and mathematical correctness are not correlated, at all. – Asaf Karagila Oct 5 '12 at 12:51
More specifically, the claim is that $\pi = 2 + \frac{2}{\sqrt{3}}$. – Nate Eldredge Oct 5 '12 at 12:55

2 Answers

The triangle he refers to does not have one half of the area of the circle, as he claims. This is where the error comes from.

In the future, I might recommend to not bother reading sites that claim obviously false things, like $\pi=3.15...$.

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Thanks for you answer. I will accept it if you prove the claim The triangle he refers to does not have one half of the area of the triangle, as he claims together with @picakhu question. I agree with the last paragraph, this is just a little puzzle not a research line. – pion Sep 18 '12 at 12:34
The author of the web page does not proof his claim that the areas are the same. The disproof is simple: If the author of the web page were right, we would have $\pi>3.15$. However we know from established proofs that $\pi<3.15$. Therefore the author's unproven claim cannot be true. – celtschk Sep 18 '12 at 12:44
@celtschk Thanks for your comment. What "established proofs" are you referring to? – pion Sep 18 '12 at 13:12
@pion: For example the proof that $\pi<22/7$. Since it is easily verified that $22/7<3.15$, this also proves that $\pi<3.15$. – celtschk Sep 18 '12 at 16:20

I'm not sure he is trying to say that his estimate of $\pi$ is better, just that this construction provides a reasonably good estimate. And it does, but we know this only because we know a good approximation of $\pi$ beforehand (that is, we can compare the calculated value to 3.1415926...)

His error estimate makes me wonder though, since it's way off. If he is trying to claim that his estimate of $\pi$ is better than the usual 3.1415926..., then he would need to better explain his error estimate, which comes out of nowhere.

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it is pretty clear that he is claiming that his estimate is better. Look here - dinbali.com/?page_id=78 – picakhu Sep 18 '12 at 12:41
The construction $22/7$ provides an even better estimate and takes somewhat less work :P he is just a crackpot who thinks he has trisected the angle. – Ben Millwood Sep 18 '12 at 13:34
@BenMillwood Not only trisecting the angle: Over fifty years of research, he has achieved solving the classic problems of mathematics such as trisecting the angle, squaring the circle, duplicating the cube, constructing the n-sided regular polygons, deriving the genuine value of pi ;) – pion Sep 18 '12 at 13:51
Galois must be turning in his grave now... – Johannes Kloos Sep 18 '12 at 14:24
@BenMillwood Didn't Archimedes get 22/7 by calculating the areas of inscribed and circumscribed 96-gons? That sounds like a lot of work to me... Unless I'm mistaken about the origin of 22/7. I'm honestly not willing to look too closely into the rest of this crackpot's website though to see what lengths he goes through and how much "work" it actually is to obtain his estimate of $\pi$. – Richard Sullivan Sep 18 '12 at 14:34