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I have read about how they calculate the value of $\pi$ using polygons.

A hexagon inside a circle gives an approximation of $\mathbf{\pi} = 3.00$, then doubling the number of the number of polygon sides gives a closer approximation of $3.10582$, and if we keep doubling the amount of sides the value for $\pi$ starts to plateau. At $1536$ sides, we get the first six digits of $\pi = 3.14159$.

calculating pi spreadsheet

Would it be fair to say that the value of $\pi$ that we use is just a very close approximation? To me it seems that no matter how many sides a polygon has, its perimeter will always be less than the circumference of a circle that encloses it.

Are there any other proofs for $\pi$ that are 'not too technical'?

Folding Circles
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    You can also use circumscribed polygons and take an arithmetic mean of the two results. Does this sound more convincing? – Yuriy S Jan 29 '18 at 00:18
  • Related: https://math.stackexchange.com/questions/1295373/computing-irrational-numbers/1295389#1295389 – Ethan Bolker Jan 29 '18 at 00:18
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    any practical experiments done that prove the value of π What does that even mean? See perhaps Buffon's needle. – dxiv Jan 29 '18 at 00:18
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    Circles do not exist in the physical world, so what kind of practical experiments do you want? – Yuriy S Jan 29 '18 at 00:19
  • a bubble is a circle, perfect sphere. Practical experiments, maybe using pulsed light and a rotating circle. Like they do when checking the timing for an engine.. – Folding Circles Jan 29 '18 at 00:22
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    A bubble is not a perfect sphere by any means – Yuriy S Jan 29 '18 at 00:27
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    Look at this interesting site the world of PI for a history about the calculations / algorithms etc. on $\pi$ – G Cab Jan 29 '18 at 00:28
  • Is the problem the finite precision? Of course any value of $\pi$ we actually use is just an approximation. Other methods allow us to compute approximations from above, from below and even compute any digit of $\pi$ separately – Yuriy S Jan 29 '18 at 00:38
  • Yuriy - I am just trying to understand, I find it fascinating and was mainly wondering if there are other proofs that either agree or disagree with the 2 billion digits that we use. Thanks for your input. – Folding Circles Jan 29 '18 at 00:41
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    A practical experiment (in contrast to a calculation) always gives approximate values. You can never measure exactly $1$ second, $1$ meter or whatsoever because the precision is always limited. Moreover, the decimal expansion of $\pi$ does not have a period (We know that $\pi$ is irrational), so we can never write down the complete decimal expansion ( not even with marking a period like in the case of rational numbers). – Peter Jan 29 '18 at 00:42
  • Thanks peter. so perfection is unobtainable through any practical experiment. Even at the quantum level you think? – Folding Circles Jan 29 '18 at 00:48
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    It helps me to think of irrational numbers as collections of algorithms which all are proven to result in the same digits for a particular number. Then I do not have to ask myself if this infinite string of digits even exists. Because there's several (or at least one) algorithm which can give me any number of digits I want. Of course there's uncomputable numbers, but who needs them – Yuriy S Jan 29 '18 at 00:48
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    We can also safely say that there is no practical expermient (or measurement) , that can approve the first $200$ digits of $\pi$ (Probably, this is already a big overshoot). And, if we have , for example , a circle or a ball, we cannot even determine whether it is actually an exact circle, ball or whatsoever by an experiment. The polygon-method never reaches $\pi$ , but it approaches $\pi$. Archimedes determined $\pi$ astonishing precisely this way. The accuracy appears poor for today standards, but with the tools Archimedes had it was a really good performance. – Peter Jan 29 '18 at 00:49
  • Excellent, thanks guys. Yuriy, so if pi cannot be written using a fraction or even 'placed' into a geometrical shape, how does 'nature' use/store infinite numbers? We know that pi is used in the plank length. – Folding Circles Jan 29 '18 at 00:52
  • Also interesting : The very good approximation $\pi\approx \frac{355}{113}$ was found astonishing early. It is based on the continued fraction method and is an approximation sufficient for most applications (In practice, often the easier but worse approxmation $\frac{22}{7}$ is used). – Peter Jan 29 '18 at 00:54
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    Folding circles, nature doesn't store numbers anywhere. It doesn't even know about them, or mathematics. In any case, that's a philosophical question. As for Plank's length, it's an approximate number, depending on experiments, and its global significance is hypothetical – Yuriy S Jan 29 '18 at 01:01
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    @FoldingCircles Ironic, that the formulas for the area and the circumference for the "perfect shape" (the circle) contains the irrational number $\pi$ which is believed to contain every finite digit-sequence. – Peter Jan 29 '18 at 01:01
  • @peter haha, i meant 'apart from the circle' . thank you – Folding Circles Jan 29 '18 at 01:04
  • @Peter " ..the irrational number π which is believed to contain every finite digit-sequence. " From the word 'believe' I understand that this has not been proved; so is it an unproven conjecture; which seems to hold for a large number of sequences? – Reader Manifold Jan 29 '18 at 02:25
  • you can never prove it because its infinite. we can never get to the end of the number. – Folding Circles Jan 29 '18 at 02:31
  • I don't think it is an infinite sequence means that it is unprovable. Why should it be? – Reader Manifold Jan 29 '18 at 02:33
  • Similar to this recent question: https://math.stackexchange.com/questions/2623374/is-infinitely-small-error-the-same-as-0-error – Hans Lundmark Jan 29 '18 at 09:03
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    @Deepakms There might be a proof, but the actual situation is much worse. We do not even know for any digit whether it appears infinite many often in $\pi$, nor can we rule out that eventually only two distinct digits appear. The conjecture is based on the calculated digits, there is no further evidence. – Peter Jan 29 '18 at 11:24
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    Often it is argued that "almost all" real numbers are normal (implying that every finite digit sequence appears), but $\pi$ is not a "random" number, so this argument gives no additional evidence. On the other hand,to show that some finite digit sequence does NOT appear, will be almost certain impossible. – Peter Jan 29 '18 at 11:33
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    @FoldingCircles This argument is not necessarily true. We know that there are infinite many primes, although we can never calculate them all. A proof that $\pi$ is normal, very well can exist, but noone has a clue how this might work. – Peter Jan 29 '18 at 11:37
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    @Deepakms Concerning the end of your comment. It is also conjectured that every algebraic irrational number is normal to every base. Quite a courageous statement, considering that no algebraic irrational number is known to be normal to any base. Neither has such a normality been proven for $\pi$ or $e$. – Peter Jan 29 '18 at 11:46
  • @Peter can you please look at my newest post. thank you – Folding Circles Feb 18 '18 at 04:03

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