Talk:Pi

Polygons inscribed in a circle of a diameter of 1 unit:

Polygons inscribed in a circle of a diameter of 1 unit:

Let there be an equilateral triangle inscribed in a circle and the measure of the sides of the triangle is a measure of an angle of sin of 60°, as the measures of the angles of the triangles decrease by a multiple of 2 the sides of the triangles increase by a multiple of 2. From 3 sides to 6 sides to 12 sides to 24 etc... The sides of the polygons are a measure of angles with isosceles triangles in each polygons and in Isosceles triangle you only need one angle to find the lenght of the sides of the triangles since the sides correspond to the measure of an angle.

The following statement holds true for isosceles triangles:

$\cos C=1-(\ (\cos A)^{2}+\ (\cos B)^{2})$

199.7.157.97 (talk) 14:04, 14 February 2024 (UTC)

This post was correctly removed recently, then re-added. To the poster: The talk page is not for discussing the subject, or for publishing your own research on the subject -- it is for discussing changes to the article. If you suggest your text should be added, we'd need quotable independent quality sources for it. As it stands, your post should be removed again -- but I suggest, to other editors, that we leave it here for a few days, to see if something relevant comes of it. Otherwise, after that, feel free to delete my post along with the post above! Nø (talk) 14:38, 14 February 2024 (UTC)

I don't understand the direct relevance to π. Seems like a subject for regular polygon (and indeed is already more or less discussed at Regular polygon § Circumradius). –jacobolus (t) 18:47, 14 February 2024 (UTC)

If it seems less discussed, there is a reason behind it. But It should be mentioned that at least why the following statement ${\frac {\sin 60}{2x^{2}}}\times (3)(2^{x})=\pi$ is not mentioned with polygons inscribed within a circle and that is exactly what it represent. 199.7.157.97 (talk) 20:47, 14 February 2024 (UTC)

correction$\ sin{\frac {60}{2x^{2}}}\times (3)(2x^{2})$ 199.7.157.97 (talk) 21:02, 14 February 2024 (UTC)

$\sin {\frac {60}{2^{x}}}\times (3)(2^{x})$ sorry 199.7.157.97 (talk) 21:29, 14 February 2024 (UTC)

Can you be specific about what change to the article you are proposing? I don't really understand what you mean here.

Are you trying to say that you can approximate π by forming a $3\cdot 2^{n}$-gon? So far as we know this was first done by Archimedes in Measurement of a Circle. Expressed in modern notation the relevant identity is ${\textstyle \cot {\tfrac {1}{2}}\theta ={}}$$\cot \theta +\csc \theta ={}$$\cot \theta +{\sqrt {\cot ^{2}\theta +1}},$ starting from $\cot {\tfrac {1}{3}}\pi =1{\big /}{\sqrt {3}},$ which leads to $\cot {\tfrac {1}{6}}\pi ={\sqrt {3}},$ $\cot {\tfrac {1}{12}}\pi =2+{\sqrt {3}},$ $\cot {\tfrac {1}{24}}\pi =2+{\sqrt {2}}+{\sqrt {3}}+{\sqrt {6}},$ etc. Archimedes eventually came up with $3+{\tfrac {10}{71}}<\pi <3+{\tfrac {1}{7}}.$ Over the following centuries this was taken to closer approximations, e.g. by al-Kashi who computed 16 digits. This is mentioned in the section Pi § Polygon approximation era. There is a bit more detail at Approximations of π.

We could plausibly say a bit more about it, especially at our article Measurement of a Circle which is currently not very complete. A source is

Miel, George (1983). "Of calculations past and present: the Archimedean algorithm" (PDF). American Mathematical Monthly. 90 (1): 17–35. doi:10.1080/00029890.1983.11971147. JSTOR 2975687.

–jacobolus (t) 22:01, 14 February 2024 (UTC)

I don't know what measurement of unit Archie's used and with polygons inscribed in a circle there aren't any, even considering triangles angles, so pi is without an SI unit.The only measurement to pi are the measures of an angle this is where polygons kick in and Archie's discovery was a guess and he was right.But his work was not finish and many people don't agree or are not sure.For me I am sure that pi is infinite and true. 199.7.157.97 (talk) 23:26, 14 February 2024 (UTC)

Quick question. Who is "Archie"? Dedhert.Jr (talk) 05:16, 15 February 2024 (UTC)

Presumably Archimedes (mentioned in my previous comment). –jacobolus (t) 07:18, 15 February 2024 (UTC)

I don't understand "pi is infinite and true". On the other hand, Archimedes' method can be translated in modern mathematical language as the observation that the half perimeter of the regular $3\cdot 2^{n}$-gons inscribed in and circumscribed to the unit circle are respectively (angles in degrees) $3\cdot 2^{n}\sin {(60/2^{n})}$ and $3\cdot 2^{n}\tan {(60/2^{n})}.$ So

$3\cdot 2^{n}\sin {(60/2^{n})}<\pi <3\cdot 2^{n}\tan {(60/2^{n})}.$

This is a special case of $\sin x<x<\tan x$ for small angles in radians. However, for computing the above approximations of π, one needs a method for computing trigonometric functions. This is this method that is described above by Jacobulus. As this method has only a historical interest, my opinion is that there is no need to give these details in the article. D.Lazard (talk) 15:56, 15 February 2024 (UTC)

Thanks.pi can be expressed in degrees and in radian.What I meant by infinite is that there is no end to the numbers of the circumference of the circle,and by true I meant it's accurate. 199.7.157.97 (talk) 17:07, 15 February 2024 (UTC)

Please note: wikipedia is not a forum. Nø (talk) 11:20, 15 February 2024 (UTC)

Discovery and Invention of Pi

Though pi has been mentioned in many Egyptian and Babylonian civilizations' sources, pi was first said to be discovered by the Archimedes of Syracuse in Greece more than 2200 years ago around 250BC. The Chinese mathematicians approximated π to seven digits, while Indian mathematicians (especially Aryabhata during the Gupta dynasty achieved a five-digit approximation using geometrical techniques. William Jones then devised the Greek symbol π to represent pi in 1706. It was then popularized by Leonhard Euler in 1737. Georges Buffon devised a method to calculate the value of pi based on probability. The invention of calculus allowed the calculation of the digits of pi up to a hundred digits which was sufficient for scientific purposes. Aanchal.Mishra20 (talk) 04:37, 9 March 2024 (UTC)

I don't understand what you are getting at. The article already discusses all of this in greater detail. –jacobolus (t) 06:52, 9 March 2024 (UTC)

Edit request: Mistake in arctan equation

In the section History->Infinite Series there is a mistake in the equation $\pi/4=5 \arctan(1/7)+2\arctan(3/77)$. This should be replaced by the correct version used by Euler $\pi/4=5 \arctan(1/7)+2\arctan(3/79)$, i.e. 79 instead of 77 in the denominator of the second arctan. 134.2.251.3 (talk) 18:11, 14 March 2024 (UTC)

 Done Tito Omburo (talk) 18:20, 14 March 2024 (UTC)

Thank you! 2A02:3038:600:F6A4:1049:1828:E0EC:9F34 (talk) 18:25, 14 March 2024 (UTC)

Thanks. I believe I was responsible for that typo. –jacobolus (t) 18:25, 14 March 2024 (UTC)

Mistake in definition

In the definition using integral, the function of the upper half of the circle is in the denominator. It should just be int_{-1}{1}sqrt(1-x^2)dx if I’m not mistaken. Marsi Viktor (talk) 22:14, 4 April 2024 (UTC)

This is the integral for arc length of the circle, not area. Tito Omburo (talk) 22:25, 4 April 2024 (UTC)

You are right, sorry :) Marsi Viktor (talk) 08:06, 5 April 2024 (UTC)

Mistake in meandering river

I think there's an error in this paragraph:

Under ideal conditions (uniform gentle slope on a homogeneously erodible substrate), the sinuosity of a meandering river approaches π.

But if we look at Posamentier & Lehmann (2004, p. 141):

[…] We then have a sum of semicircular arcs that will be compared to a single semicircular arc with a diameter equal to the distance the full distance the river will have traveled […].

• I = length of the river from the source A to the month B
• AB = (straight) distance between the source A to the month B
• […]
• a = approximation of the river's length […]

$${\frac {\pi }{2}}={\frac {a}{AB}}={\frac {I}{AB}}$$

So, shouldn't it be "the sinuosity of a meandering river approaches π/2"? Vinickw‍✉ 12:18, 11 April 2024 (UTC)

A sinuosity of pi/2 corresponds to gluing together two semicircles into an S shape. I think the claim in the article is that an ideal meandering river is pi, which would be more sinuous than this (so the river tends to close up more, and so there can be oxbows, for example). That seems intuitively reasonable to me, and also not actually contradicted by the above cited paragraph. I think one should consult the first cited source for clarity:

Stølum, Hans-Henrik (1996), "River Meandering as a Self-Organization Process", Science, 271 (5256): 1710–1713, Bibcode:1996Sci...271.1710S, doi:10.1126/science.271.5256.1710, S2CID 19219185.

Unfortunately, I do not have access. Tito Omburo (talk) 21:10, 11 April 2024 (UTC)

Yeah, I'm was talking about it with my professor yesterday, I'm going to read this article once again in detail. By the way, I think you can access it via meta:The Wikipedia Library, you seem to meet the requirements. Vinickw‍✉ 11:50, 12 April 2024 (UTC)

That source has "In the simulations ... [t]hese opposing forces self-organize the sinuosity into a steady state around a mean value of s = 3.14, the sinuosity of a circle (π).... The mean value of π follows from the fractal geometry of the platform." I see no mention of π/2. NebY (talk) 12:14, 12 April 2024 (UTC)

Great, thanks. Related question: is this claimed to be proven, or just conjectured based on simulations? Tito Omburo (talk) 12:49, 12 April 2024 (UTC)

It's explicitly what simulations with a fluid mechanical model show. That's not a direct answer to your question, because I wouldn't talk about such a thing as river sinuosity under ideal conditions being proven or describe such modelling as conjecture. I do fear that the modelling demonstrates that Posamentier & Lehmann's theoretical approach, at least as summarised above, may not be realistic. NebY (talk) 13:10, 12 April 2024 (UTC)

Yesterday my professor (pinging him, maybe he help @Cesarb89) found some files, like this one, it says on page 10 that the value 1.5 (note that π/2 ≈ 1.57) "arbitrarily divides rivers with high sinuosity (greater than 1.5) of those with low sinuosity (less than 1.5)". A meandering river (in Portuguese: canais meandrantes) is a single channel river with high sinuosity (this is also the definition on Meander). This makes sense, after all, if we look at the image on Posamentier & Lehmann (2004), the river is still far from creating oxbow lakes. Vinickw‍✉ 16:11, 12 April 2024 (UTC)

A meandering tale: the truth about pi and rivers by James Grime[1] found an average much lower than π, and despite some outliers (5.88!) relatively close-packed data. I found some of the comments interesting: should immature rivers be excluded, should a meandering river's length be measured with respect to the downhill direction(s), and is it a version of the coastline problem?

Perhaps, rather than our current confident statement that sinuosity approaches π, we should say that various attempts have been made to relate sinuosity to π. NebY (talk) 17:49, 12 April 2024 (UTC)

This is a huge finding. It's important to note that Stølum (1996) uses a simulation of rivers, so it's reasonable to assume that real-world conditions may yield different results. Although pimeariver.com is no longer active, the latest archive, from 31 May 2019, shows that the average of 280 rivers (22 more than what's written on the Guardian) is 1.916, still far from π, and the value is moving away from π/φ. Vinickw‍✉ 19:41, 12 April 2024 (UTC)

I would imagine that the steepness of slope makes a huge difference, and probably also the local geology, type/quantity of plant cover, amount of rainfall, seasonal variation in water quantity, etc.

I bet if you look up sources about hydrology / hydrographic engineering there is probably more detailed/careful technical material than in sources about mathematics per se. –jacobolus (t) 00:46, 13 April 2024 (UTC)

yeah, I'm a little concerned about the way our article approaches this, making it seem much more definitive. This is why I wondered to what extent there is something like a "theorem" as opposed to "someone ran a simulation once". It seems like in-text attribution would be warranted. Tito Omburo (talk) 15:46, 13 April 2024 (UTC)

Perhaps something along these lines?

Analyses of river sinuosity (length relative to distance) have found it to approach π, π/2 and neither.

Given that there are so few studies of any relationship between sinuosity and pi (though I see Stølum has published a little more) and that the results are so varied, I'm not sure our article should give more space to the idea. NebY (talk) 16:08, 13 April 2024 (UTC)

I agree that it is unfortunately too tenuous, and should be removed. Tito Omburo (talk) 09:45, 15 April 2024 (UTC)

You're right, simple removal's better than dwelling on the claim and its contradictions.  Done NebY (talk) 10:30, 15 April 2024 (UTC)

References