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The definition of pi assumes that the diameter/circumference ratio is the same for all circles. Is this something that has ever been proved? It seems a self-evident truth, but are there such things in mathematics? Esedowns (talk) 12:35, 2 September 2023 (UTC)Reply[reply]

It's not self-evident, relevant or even true. What is true is that in a Euclidean space the ratio is invariant. I updated the lead to reflect this. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:08, 3 September 2023 (UTC)Reply[reply]

If not explicitly specified otherwise, circles are those of Euclidean geometry. D.Lazard (talk) 17:04, 3 September 2023 (UTC)Reply[reply]

That's certainly the common case, but in an article about Mathematics there should be no such assumption. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:34, 5 September 2023 (UTC)Reply[reply]

In an article aimed at a general audience, unqualified "circle" means a flat Euclidean circle, unless something else is obvious from context. If there's some ambiguity because e.g. Euclidean vs. spherical circles are being directly compared, then it can be directly specified for clarity.

@Esedowns to answer your question plainly: yes it has been proved that the ratio of the circumference to diameter of a circle is the same for all circles. But as @Chatul points out, if you generalize the concept of "circle" to apply in contexts other than Euclidean geometry, then π can take another value, can vary from circle to circle, can be infinite, or can be undefined. –jacobolus (t) 12:52, 5 September 2023 (UTC)Reply[reply]

Many thanks for your reply. I wonder where the proof is. Something I read years ago, by Russell I think, said there are begged questions in Euclid.

Esedowns (talk) 15:08, 5 September 2023 (UTC)Reply[reply]

Euclid proves that the ratio of area of a circle to a square on its diameter is constant in Elements 12.2 (c. 300 BC). Archimedes' Measurement of a Circle (c. 250 BC) shows that area of a circle is half the area of the rectangle of sides radius · circumference. The combination of these two results may be what you are looking for. Modern mathematicians might disagree about whether the methods used were sufficiently rigorous. Nowadays concepts like arc length are formalized using calculus. –jacobolus (t) 17:55, 5 September 2023 (UTC)Reply[reply]

Many thanks again. It is all above me really! Esedowns (talk) 18:33, 5 September 2023 (UTC)Reply[reply]

A literature search turns up:

Richeson, David (2015). "Circular Reasoning: Who First Proved That C Divided by d Is a Constant?". The College Mathematics Journal. 46 (3): 162–171.

Lima, Fábio M. S.; Jordão, Pedro G. F. (2022). "Why is it that the Ratio of Any Circle's Circumference to its Diameter is a Constant?". The College Mathematics Journal. 53 (3): 171–182.

–jacobolus (t) 18:31, 5 September 2023 (UTC)Reply[reply]

Why is the father of Egyptology, Flinders Petrie, labelled a "pyramidologist"? Surely that is a serious insult? 165.73.112.52 (talk) 05:49, 6 September 2023 (UTC)Reply[reply]

What I gleaned in a quick skim through the source (Roger Herz-Fischler's The Shape of the Great Pyramid), is that Flinders Petrie was an adherent of the "π theory" about the proportions of the Egyptian pyramids starting at age 21, which his father had been promoting for years beforehand. He never explicitly recanted such claims, but it seems like later in his career he stopped talking about it. I don't think this is a priori insulting, but perhaps there's a more concrete way of describing this, still linking to pyramidology without calling anyone specifically out with a term that might lead readers to conclude they were a pseudoscientific crank (aside: there were plenty of other prominent, serious people who were "pyramidologists" in the 19th century). –jacobolus (t) 06:06, 6 September 2023 (UTC)Reply[reply]

The context in which we mention Flinders Petrie is that of theories that the dimensions of the great pyramid are based on π. Those theories are indeed pyramidology, and we should say so. It is accurate labeling, not insult for the sake of being pejorative. Flinders Petrie is notable for other things (including debunking some other branches of pyramidology) but it is not those other things that we are discussing here. —David Eppstein (talk) 07:30, 6 September 2023 (UTC)Reply[reply]

The dimensions of the great pyramid do indeed supply an approximation of π (twice base over height is 880/280 = 22/7). That much is agreed by scholars. The question is, was it designed with that in mind, or is it merely a consequence of the chosen slope angle?

The question is actually more subtle than that. Khufu is based on a whole-cubit Kepler triangle, while Khafre is based on the Pythagorean 3:4:5 triangle. Thus they form a matching pair of the two unique right-angled triangles (arithmetic sequence 3:4:5 and geometric sequence 1:√φ:φ). Egyptologists dismiss this as "co-incidence" because they don't want to answer questions about Pythagorean triangles and the golden ratio. (Can't see the wood for the trees!)

The π ratio is a side effect of the Kepler design, since 4/π ≈ √φ. Both the φ and π values are within 0.1% of true. So good approximations for a stone building.

I think WP's definition of "pyramidologist" is too wide ... it effectively shuts down much discussion about all the mathematics on display at Giza (and elsewhere)... the motivation being because such analysis effectively challenges the accepted time line. This is the "settled science" argument which is anathema to any sort of progress in understanding the past.

Calling people names is not the way to deal with bad arguments. Play the ball, not the man.

As for Petrie, he went to Egypt to disprove assorted theories, and devotes a whole chapter to debunking, as well as in various other places in the text. But Petrie was very smart, he could see that there were things about Giza that did not mesh with the known history. He hints at these carefully. 165.73.112.52 (talk) 15:47, 10 September 2023 (UTC)Reply[reply]

"The context in which we mention Flinders Petrie is that of theories that the dimensions of the great pyramid are based on π. "

Can you prove that they are not? 165.73.112.52 (talk) 15:58, 10 September 2023 (UTC)Reply[reply]

@David Eppstein, inre "we should say so" – the problem with our current presentation is that this is a single throwaway line. The article here doesn't do any kind of deep analysis of pyramidology or even the "π theory", explaining full context about the characters involved or their specific claims. Instead we gratuitously call out Petrie specifically, attach a label that could plausibly lead readers to conclude he was a crank, and then leave the subject aside. Personally I don't think it would materially affect our presentation to say "some 19th century pyramidologists ..." without specifically naming anyone. But if you think we need an example name to make the claim more concrete, we should pick whoever originated or most forcefully propounded the π theory. –jacobolus (t) 16:30, 10 September 2023 (UTC)Reply[reply]

I'd be ok with omitting Petrie's name here. —David Eppstein (talk) 17:07, 10 September 2023 (UTC)Reply[reply]

Ridiculous. Petrie is the father of Egyptology. To call him a pyramidologist is a total insult. He is the one who fought the pyramidologists, headlined by Scottish astronomer C Piazzi Smyth. This battle is famous and Petrie is credited with debunking the pyramidologists and putting Egyptology onto forensic footings. Remove his name and replace it with a real pyramidologist. John Taylor in his 1859 book started the Pi theory and Piazzi Smyth and the others went with that. Get Petrie's name off. It's simple or you are remaining willfully ignorant of the facts and of Petrie's high esteem among Egyptologists today. He is the patriarch. EulerConstant (talk) 15:20, 10 September 2023 (UTC)Reply[reply]

"s" missing. KDAM71 (talk) 20:24, 8 October 2023 (UTC)Reply[reply]

Fixed. Thanks for alerting us. HiLo48 (talk) 05:40, 9 October 2023 (UTC)Reply[reply]

This edit request has been answered. Set the |answered= or |ans= parameter to no to reactivate your request.

change

"It is a transcendental number, meaning that it cannot be a solution of an equation involving only sums, products, powers, and integers"

to

"It is a transcendental number, meaning that it cannot be a solution of an equation involving only finite sums, products, powers, and integers"

the word finite is necessary here, because pi *can*, in fact, as is described in the sections below, be expressed as an infinite sum. 188.167.159.122 (talk) 17:42, 10 October 2023 (UTC)Reply[reply]

 Done Tito Omburo (talk) 18:30, 10 October 2023 (UTC)Reply[reply]

I have looked ad more digits: 3.1415926535927 66.58.219.246 (talk) 17:38, 21 October 2023 (UTC)Reply[reply]

Pi § Approximate value and digits lists 50 digits and links to OEIS where you can see many more (here are the first 20,000 digits). There are also many places on the internet where you can find as many digits as you care to read (here are the first 1,000,000 digits and here are the first 100,000,000,000,000 digits), as well as computer code to generate arbitrarily many on your own computer. –jacobolus (t) 18:05, 21 October 2023 (UTC)Reply[reply]

This edit request has been answered. Set the |answered= or |ans= parameter to no to reactivate your request.

I would like to include the first 314 decimals of Pi, due to the first numbers in pi being 3, 1 and 4. I don't know, the idea just came to my head. Here's the source to obtain the numbers: https://3.141592653.com JonJohnJimmy (talk) 18:52, 14 November 2023 (UTC)Reply[reply]

I don't think that's a particularly good reason for including this content. —David Eppstein (talk) 19:09, 14 November 2023 (UTC)Reply[reply]

Towards the end of the lead we say:

In modern mathematical analysis, it is often instead defined without any reference to geometry; therefore, it also appears in areas having little to do with geometry, such as number theory and statistics.

That therefore makes no sense. Rather,

Because it appears in areas having little to do with geometry, such as number theory and statistics, it is possible to define it without any reference to geometry, as is usually done in modern mathematical analysis.

However, I will not make that change; I hope someone else can come up with a better wording. Nø (talk) 08:42, 8 December 2023 (UTC)Reply[reply]

In that context, because makes as little sense as therefore. How about Because it appears in areas having little to do with geometry, such as number theory and statistics, it is convenient to define it without any reference to geometry, as is usually done in modern mathematical analysis.? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:49, 8 December 2023 (UTC)Reply[reply]

I've edited it using "and" rather than asserting causation in either direction; the body of the article doesn't, as far as I can tell, assert one caused the other so our WP:LEAD summary shouldn't. I couldn't see that "often" was supported by the body of the article so I dropped that too, rather than replace it with "usually". NebY (talk) 14:49, 8 December 2023 (UTC)Reply[reply]

Good solution. Tito Omburo (talk) 16:16, 8 December 2023 (UTC)Reply[reply]

This 'often' or 'usually' is probably a supportable claim, but "can be defined" also seems fine. –jacobolus (t) 19:07, 8 December 2023 (UTC)Reply[reply]

Yes indeed, quite possibly supportable but maybe not worth the trouble? Thanks, both, and to Nø for spotting the problem. NebY (talk) 17:56, 9 December 2023 (UTC)Reply[reply]

I expect the "pop culture" bit to be full of trivia, but the claim in Carl Sagan's novel seems flaky. I am not an expert, but the nature of the digits of pi suggests that there is probably a proof that given any message (or its logical converse), it is indeed encoded somewhere in the digits of pi. I removed the following sentence:

In particular, when Moses asks God for His name, God replies, "I Am who I Am." in the 3rd chapter, 14 and 15 lines^[1] of the Book of Exodus (${\displaystyle \pi \approx 3.1415\dots }$).

Meaning is not very clear, but for (almost?) every book of the bible there is a 3rd chapter, which includes verses 14 and 15, which remarkably enough are always consecutive. Imaginatorium (talk) 11:13, 9 December 2023 (UTC)Reply[reply]

Thanks for removing that. I'd just as soon take out the whole paragraph about the novel, song, and TV episode. –jacobolus (t) 17:23, 9 December 2023 (UTC)Reply[reply]

That is a plot point in the novel Contact. So, it is the kind of thing we can mention if we have secondary sourcing that indicates it is significant enough to merit inclusion. Whether it's mathematically legitimate is a separate question. XOR'easter (talk) 18:54, 9 December 2023 (UTC)Reply[reply]

In Carl Sagan's 1985 novel Contact it is suggested that the creator of the universe buried a message deep within the digits of π. This suggestion is supported by the Bible. When Moses asks the creator for His name, the creator replies, "I Am who I Am." in the 3rd chapter, 14 and 15 lines^[2] of the Book of Exodus (${\displaystyle \pi \approx 3.1415\dots }$). Why cannot we add this information to support this valuable Carl Sagan's suggestion to this article? Or perhaps we should remove Carl Sagan's suggestion from this article as it is unsupported? Guswen (talk) 22:21, 9 December 2023 (UTC)Reply[reply]

This is discussed above.

• The existence of a third chapter with verses 14 and 15 consecutive to each other is trivial (true of anything divided into sufficiently many chapters and verses) and the contents of that chapter are not particularly relevant, so this does not actually support the suggestion.
• We cannot include this material without secondary published sources discussing it. Without these sources, the idea that it has some connection to hidden messages in pi is original research.

—David Eppstein (talk) 22:43, 9 December 2023 (UTC)Reply[reply]

The Bible numerology is clearly un-encyclopedic nonsense. That Carl Sagan's novel included π as a plot point is not controversial, and there are secondary sources discussing it. The only relevant question about it is whether it's important enough to include in this article, or whether it seems too trivial. –jacobolus (t) 23:09, 9 December 2023 (UTC)Reply[reply]

Also, "3.1415" is not "deep within the digits of pi", so even if we include the material about Sagan, its connection to the Bible verse that Guswen wants to add seems extremely tenuous. —David Eppstein (talk) 23:14, 9 December 2023 (UTC)Reply[reply]

Thank you for this feedback, gentlemen. I will think about it. For now, let me just say that for the man in the street ${\displaystyle \pi \approx 3.14\dots }$ not even ${\displaystyle \pi \approx 3.1415\dots }$, so the 3rd chapter 15 line of the Book of Exodus suffice. The 15th line is redundant.

And I don't know much about those "deep within the digits of pi". But perhaps we should create a Wikipedia article within the digits of pi to cover this seemingly overlooked issue (?) Guswen (talk) 00:08, 10 December 2023 (UTC)Reply[reply]

I don't understand what your complaint is. The removed material should not be included in Wikipedia. An article within the digits of pi does not seem worthwhile to create. Newly invented numerology about the Bible and π does not have any encyclopedic value (cf. Wikipedia:No original research). Old, well sourced numerology with some attested historical significance may be relevant at the page biblical numerology, but is not relevant at pi. –jacobolus (t) 02:26, 10 December 2023 (UTC)Reply[reply]

But I did not "invent" any numerology. This numerology has been known at least since the 6th century BCE, when the Book of Exodus was written, according to modern scholars.

Carl Sagan's suggestion is valid and reveals historically the first correct approximation of ${\displaystyle \pi \approx 3.1415}$. Earlier approximations of π dated around 1650 BC found in Egypt gave the value of π as ${\textstyle \pi \approx {\bigl (}{\tfrac {16}{9)){\bigr )}^{2}\approx 3.16}$, for example.

Carl Sagan's suggested that "the creator of the universe buried a message deep within the digits of π". But what could be the content of such a message (according to Carl Sagan) if not the name of the creator of the universe? And this name is revealed in the 3rd chapter, 14,15 lines of the book of Book of Exodus.

Many books contain 3rd chapters with lines 14 and 15, but those lines do not carry any message from the creator of the universe, as Sagan suggested.

Guswen (talk) 08:49, 10 December 2023 (UTC)Reply[reply]

Please stop writing garbage. (After all, you claim to have a degree in mathematics.) FWIW, John 3: 14, 15 also contains a "message from the creator of the universe", viz. blah blah, sorry, look it up yourself. What could be the content of such a message (according to you) if not something about "eternal life"? Sagan is (obviously) referring to the idea that peering inside the digits of pi there would appear a "message" encoded somehow in the digits. A message that was not known beforehand. Not that an index to the bible (the chapter and verse numbers, which were in any case added much later) would somehow point to a particular message. I pointed out above that because of the infinite elasticity allowed to interpreters of such claims, it is always going to be possible to find any desired messages if determined enough: for example "Donald Trump savio(u)r" or "Donald Trump harbinger of doom"; WP does not require editors to have any mathematical sophistication whatsoever, but you cannot claim have any and still believe this nonsense. Anyway, tactically, the discussion here is over. Imaginatorium (talk) 09:56, 10 December 2023 (UTC)Reply[reply]

I do not need my degree in mathematics to see that ${\displaystyle \pi \approx 3.1415}$. But thank you for the hint! Indeed, the Creator's message in John 3:14,15 (14 Just as Moses lifted up the snake in the wilderness, so the Son of Man must be lifted up, 15 that everyone who believes may have eternal life in him.) relates to His earlier message in Exodus 3:14,15. So Carl Sagan's intuition was outstanding!

"We can judge our progress by the courage of our questions and the depths of our answers, our willingness to embrace what is true rather than what feels good." Carl Sagan.

Guswen (talk) 10:31, 10 December 2023 (UTC)Reply[reply]

As far as I can tell having not read it, Carl Sagan book has nothing to do with any line 3:14–15 of the Bible. "since the 6th century BCE, when the Book of Exodus was written, according to modern scholars" – If you ever want to add this to some other page where it's relevant (e.g. a page about examples of pseudoscientific nonsense), please carefully cite these modern scholars in the discussion there. In my opinion biblical numerology is not relevant or appropriate at the page about pi the mathematical constant, and this conversation has become largely off topic for this discussion page. –jacobolus (t) 16:49, 10 December 2023 (UTC)Reply[reply]

I've read it. You are correct in saying that it has nothing to do with any line 3:14–15 of the Bible. XOR'easter (talk) 18:32, 10 December 2023 (UTC)Reply[reply]

If Carl Sagan discovered that the creator of the universe buried a message about His name and eternal life within the first five digits of π ([Ex 3.14.15], [J 3.14.15]), Sagan would have certainly written it in his book.

Unfortunately, he had not (He was agnostic. Agnostics have no motivation to read the Bible). However, he still had an outstanding intuition. And intuition is the only real valuable thing, according to your Master (Albert Einstein).

Guswen (talk) 21:46, 11 December 2023 (UTC)Reply[reply]

@Guswen: Wikipedia talk pages are not social media sites or fora for general conversation, and conversations about your religion are all off topic here. –jacobolus (t) 15:30, 12 December 2023 (UTC)Reply[reply]

Of course the claim in the Sagan novel is flaky, but the statement that the novel makes that claim is objectively verifiable and is clearly relevant to the popular culture section. That's a claim about the book, not a claim about the actual provenance of ${\displaystyle \pi }$. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:05, 12 December 2023 (UTC)Reply[reply]

It's not entirely off topic, but I still maintain it's not really helpful or meaningful to readers looking to learn about π, or even "π in popular culture". This kind of topic is better to leave at the page about the book; Wikipedia doesn't need to cross-reference every time any subject is included in any novel. I'm sure if we hunted we could find a longer list of books/poems/songs/TV dramas/documentaries/... involving π, but such a list is pretty much trivia. In any event, the book is currently mentioned, and nobody seems to be trying too hard to remove it. –jacobolus (t) 15:30, 12 December 2023 (UTC)Reply[reply]

Well, I am trying hard to remove it as an unsupported suggestion. If "the creator of the universe buried a message deep within the digits of π", as Sagan suggests, then we should inform the reader that this suggestion is supported by the Bible, as the creator of the universe buried a message about His name and eternal life already within the first five digits of π.

Otherwise, Sagan's suggestion is, indeed, a big Woo.

This leaves us with three options: 1. Remove Sagan's suggestion (and information about his book and movie) from this page, 2. Hunt for all books/poems/songs/TV dramas/documentaries/... involving π, as jacobolus proposes, and list them on this page in the "Popular Culture" section and 3. Give a reader a hint that they can find support for Sagan's suggestion, at least in the Bible (I'm a Catholic, but perhaps a message from the creator of the universe can be found in Quran as well).

Guswen (talk) 20:15, 12 December 2023 (UTC)Reply[reply]

The claim made in the article is only that Sagan used this as a plot point in his book, and it is discussed here an example of the appearance of π in popular culture. No claim is being made about the truth of Sagan's plot device, which is obviously pseudoscientific nonsense. Whether Sagan personally believed it to be true (unlikely) is irrelevant. The current text does not need to be validated by evidence external to the book, beyond secondary sources noting this as a culturally important example of π appearing in a novel. The stuff about particular lines 3:14 in the Bible has literally nothing to do with Sagan's book. –jacobolus (t) 20:20, 12 December 2023 (UTC)Reply[reply]

Indeed, there is a message from the creator of the universe also in Quran. 3rd Surah Al Imran, verses 14 and 15 ("The faithful, their character and reward") state that (Dr. Mustafa Khattab translation):

(14) "The enjoyment of worldly desires—women, children, treasures of gold and silver, fine horses, cattle, and fertile land—has been made appealing to people. These are the pleasures of this worldly life, but with Allah is the finest destination."

(15) "Say, O Prophet, 'Shall I inform you of what is better than all of this? Those mindful of Allah will have Gardens with their Lord under which rivers flow, to stay there forever, and pure spouses, along with Allah’s pleasure.' And Allah is All-Seeing of His servants,"

Therefore, Sagan had an outstanding intuition.

Guswen (talk) 20:37, 12 December 2023 (UTC)Reply[reply]

Please stop using this talk page as a forum for your irrelevant nonsense. It is off topic for this talk page, which can only be about properly sourced content about π. —David Eppstein (talk) 21:28, 12 December 2023 (UTC)Reply[reply]

Do you criticize the noble Quran Eppstein?

Guswen (talk) 22:58, 12 December 2023 (UTC)Reply[reply]

Please stay on topic. XOR'easter (talk) 23:53, 12 December 2023 (UTC)Reply[reply]

Do you criticize the Bible XOREaster?

Guswen (talk) 01:23, 13 December 2023 (UTC)Reply[reply]

@Guswen If you persist I am going to ask that you be temporarily blocked from editing. You are making a disruptive nuisance of yourself. –jacobolus (t) 01:40, 13 December 2023 (UTC)Reply[reply]

I'm aware of that.

A Christian cannot be afraid of losing face. The Son of God did this in the most shameful way.

Will you ask to temporarily block me from editing, for this entry too?

Guswen (talk) 22:24, 13 December 2023 (UTC)Reply[reply]

Ok, see WP:ANI#Guswen on Talk:Pi —David Eppstein (talk) 23:51, 13 December 2023 (UTC)Reply[reply]

The Bible didn't get chapter and verse numbers until much later --the Medieval Age. See "Chapters and verses of the Bible". These additions are just arbitrary organizational aids for the reader and are not considered canonical by major Jewish and Christian authorities . This stuff is not Biblical numerology. If the Almighty left us a message in the digits of pi, it's not a list of Biblical verse numbers.--A. B. ^{(talk • contribs • global count)} 04:55, 15 December 2023 (UTC)Reply[reply]

Another thing to keep in mind is that the statement that pi is "approximately equal to 3.14159" is accurate to five decimal places. To be the most accurate to four decimal places, it's not 3.1415, it's 3.1416. ←Baseball Bugs ^{What's up, Doc?} carrots→ 10:20, 15 December 2023 (UTC)Reply[reply]

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:

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

199.7.157.97 (talk) 14:04, 14 February 2024 (UTC)Reply[reply]

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)Reply[reply]

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)Reply[reply]

If it seems less discussed, there is a reason behind it. But It should be mentioned that at least why the following statement ${\displaystyle {\frac {\sin 60}{2x^{2))}\times (3)(2^{x})=\pi }$ 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)Reply[reply]

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

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

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 ${\displaystyle 3\cdot 2^{n))$-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 ={))$${\displaystyle \cot \theta +\csc \theta ={))$${\displaystyle \cot \theta +{\sqrt {\cot ^{2}\theta +1)),}$ starting from ${\displaystyle \cot {\tfrac {1}{3))\pi =1{\big /}{\sqrt {3)),}$ which leads to ${\displaystyle \cot {\tfrac {1}{6))\pi ={\sqrt {3)),}$ ${\displaystyle \cot {\tfrac {1}{12))\pi =2+{\sqrt {3)),}$ ${\displaystyle \cot {\tfrac {1}{24))\pi =2+{\sqrt {2))+{\sqrt {3))+{\sqrt {6)),}$ etc. Archimedes eventually came up with ${\displaystyle 3+{\tfrac {10}{71))<\pi <3+{\tfrac {1}{7)).}$ 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)Reply[reply]

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)Reply[reply]

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

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

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 ${\displaystyle 3\cdot 2^{n))$-gons inscribed in and circumscribed to the unit circle are respectively (angles in degrees) ${\displaystyle 3\cdot 2^{n}\sin {(60/2^{n})))$ and ${\displaystyle 3\cdot 2^{n}\tan {(60/2^{n})}.}$ So

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

This is a special case of ${\displaystyle \sin x<x<\tan x}$ 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)Reply[reply]

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)Reply[reply]

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