Talk:Squaring the circle

The illustration

You cannot say that they are the same area; the very point of the article is that that would require ascribing a finite value to pi (more precisely, the square root of pi). It is misleading and presumptive to put in the caption that they have the same area value. The graphic could also stand with no caption at all.

ps If you still think that the image shows a circle and square of the same area then copy it to your computer and zoom in on it. There is actually no circle in the image at all.

--Justanother 14:04, 22 August 2006 (UTC)

You are confused. In the first place, it is not claimed that illustrations are exact. They never are. But they convey ideas well. In the second place, the value of π is indeed finite, and so is its square root; if you think otherwise, you're very very confused. Perhaps you mean that its decimal expansion is only finitely long (in popular confusions, that seems to matter). Michael Hardy 18:07, 22 August 2006 (UTC)

I have to confess, I've sometimes wondered if the people (is it two of them now?) who have expressed objections of this kind to this illustration, are under the impression that the impossibility of squaring the circle means that a square and a circle can never have the same area? That's not actually what it says; it just says you can't do the ruler-and-compass construction. Michael Hardy 21:41, 23 August 2006 (UTC)

I said finite when I should have said constructible. My bad. From pi "An important consequence of the transcendence of π is the fact that it is not constructible." My point is that that caption is a lie. Do you argue that point with me?? You say "In the first place, it is not claimed that illustrations are exact.". But doesn't "A square and circle with the same area." make, for all intents, that exact claim? Why bother with that caption. Do you think that you can create a squared circle with pixels? I doubt it. This is not about mathematics, it is about whether a caption in a lie or not.--Justanother 15:54, 24 August 2006 (UTC)

An image is not the same as the object it depicts. The image presently discussed is not a square and a circle with the same area; it depicts a square and a circle with the same area, subject to the limitations of its format. But every image is limited in precision by its format; this is an inherent property of images. By your reasoning, every image caption in Wikipedia should begin with "Illustration of..." or "Photograph of...". Or perhaps, "A rectangular array of pixels making up an approximate representation of...". Fredrik Johansson 16:22, 24 August 2006 (UTC)

Hi. The problem with your analogies is that this particular article is about the possibility, or impossibility, of physically constructing (presenting) a circle and square of the same area. To then present a circle and and square and caption "whoop, there it is" is misleading. I see this as a special case in that the medium is very much the message. Here is a decent analogy: Suppose there is an article about the impossibility of capturing the image of a spirit (ghost) on film and someone add a photoshopped pic of just that and labels it "Spirit captured on film". I wouldn't be too happy about that one either. That is the real extension of my reasoning. --Justanother 14:16, 25 August 2006 (UTC)

You can represent the square and circle of equal area exactly in a computer and render them from that representation. Although I doubt this image was rendered from such a representation, there is nothing impossible about it. Whether the construction is physically possible has nothing to do with it. It is also impossible to physically construct a straight line, but I'm not seeing any complaints about that. Fredrik Johansson 15:08, 25 August 2006 (UTC)

Help me out here. How would you "represent" a square and circle of equal area in a computer and I don't mean simply storing or generating the equations. That is no different then writing them down on a piece of paper and even I can do that.--Justanother 16:43, 25 August 2006 (UTC)

That's what you do. Use a computer algebra system that can work symbolically with series expansions, and you can easily represent pi exactly and do calculations with it. Yes, you can do that too (you can do everything a computer does, except perhaps slower), though the arithmetic involved in rendering the circle based on the exact representation (write down exactly how large a part of pixel (x,y) is covered by the circle and then round that to an even number of 255ths for each x and y) is going to be so tedious that you'll wish you did use a computer, and I'll not stay up waiting for you to finish. :-) Henning Makholm 18:09, 25 August 2006 (UTC)

My points are:

• It is indeed possible for a square and a circle to have the same area (the impossibility asserted by the theorem is not that that is impossible, but rather that the rule-and-compass construction is impossible.

• Everybody knows that illustrations in geometry articles are ALWAYS approximations, whether made with pixels or with ink on paper. A theorem of geometry may say (paraphrasing) "This square has the same area as that rectangle", and accompany it with an illustration. The square and the rectangle as abstract mathematical objects do have EXACTLY the same area, and the square and the rectangle in the illustration in the book are approximations. Everyone realizes that they're obviously always approximations, so it is not a lie to say they have the same area. The assertion that two things have the same area is naturally understood to refer, not to the physical illustration, but to the abstract mathematical objects that they illustrate.

Michael Hardy 03:01, 25 August 2006 (UTC)

Hi. I appreciate your point of view. I am not a mathematician and I actually came to the article by way of the Timecube, which was referenced in another article I was reading. I had never encountered the "squaring the circle" and I found it interesting. The more we discuss, the more interesting I find it. My conclusion is while a square and a circle can, in theory, have the same area, there is NO way to represent that in the physical universe, not with ink or pixels nor with molecules or atoms or subatomic particles or whatever. That is pretty cool to me and I found that the caption detracted from my feeling of wonder. I think the simple caption "Squaring the circle" serves well. --Justanother 14:06, 25 August 2006 (UTC)

Your conclusion is wrong. The impossibility of squaring the circle is not a statement about the physical universe. It is a statement of a particular model of certain aspects of the physical universe, namely the model of "ruler-and-compass Euclidean geometry with no implicit continuity assumptions". In that model there are no circle-square pairs with equal areas, which mathematicians and physicists consider a deficiency of the model, i.e. the model is wrong. Indeed, what most phycisists and quite a lot of mathematicians will think of when you say "Euclidean geometry" is not that wrong model, but another one, namely "R² with the Euclidean metric". And there all circles have equal-area squares, easily. The model in which circles can be squared (which is the one the image seeks to illustrate) is universally considered a better fit with the physical universe than the one where they cannot. (However, it is not an exact match: General Relativity says it is incorrect, and if you want to be completely anal about it circles cannot be squared in the GR universe, simply because a perfect square is an impossible figure in GR, except - perhaps - in extremely extraordinary times and places). Henning Makholm 15:06, 25 August 2006 (UTC)

Wrong again, huh? My conclusion was "my conclusion" and I still like even if I cannot intelligently discuss theoretical physics with you. Though perhaps I should say NO way to physically represent it since that is what I meant, and that is what I took the original attempt with ruler-and-compass to be a subset of. But perhaps I am completely missing what mathemeticians love most about the problem and trivializing it for you. Sorry, then. But I certainly think that attributing to the image that it seeks to illustrate an alternate model where you can draw a squared circle is a bit of a stretch. Actually, quite a large stretch.--Justanother 16:43, 25 August 2006 (UTC)

Not a stretch at all. Illustrating a circle with the same area as a square is piece of cake, even though the illustration is necessarily approximate in pixels. That concept exists in the Euclidean geometry model, but not in compass-and-straightedge geometry model, which is really the point of the whole concept of "squaring the circle" and this article. Dicklyon 17:41, 25 August 2006 (UTC)

To Justanother: I see now that I may have misunderstood what you were trying to say. Apologies. I still think you are wrong, but in a different way: when you say that one cannot physically represent the circle you are forgetting that the word "represent" implicitly says that there is an idealization taking place: the representation will always be cruder than what it represents. So a roughish circle and square drawn in freehand with charcoal on a bumpy wall can quite well represent the mathematical perfection of a circle being squared. If you want to have the circle as a mathematically perfect and tangible object, with no representation going on, of course you can't: There will always be bumps and fuzziness at the atomic level no matter how well you trim and polish it. But that is independent of whether you intend to square that circle or not. Henning Makholm 17:58, 25 August 2006 (UTC)

I think perhaps we see here the difference in mindset between the mathematician and the engineer (smile). First, let's get representation out of the way. Yes, anything can represent anything else. That, after all, is the basis of language; that we can represent things and we can share a set of representations. So let me strike any use of the word representation on my part when I simply meant construction. So "My conclusion is while a square and a circle can, in theory, have the same area, there is NO way to construct that in the physical universe, not with ink or pixels nor with molecules or atoms or subatomic particles or whatever." So the point I make about the illustration is the one I brought up previously regarding this being a special case where the caption on a representation may imply that it is a construction.--Justanother 20:52, 25 August 2006 (UTC)

It seems that you are setting your criteria for "constructing in the physical universe" so narrowly that the physical universe can contain no circles at all. That is fine, but has nothing to do with squaring those nonexisting circles. However, for any reasonable sense of "exist" that allows any (Euclidean) circle to exist physically, it holds that a square with the same area can also exist. Henning Makholm 21:35, 25 August 2006 (UTC)

Hmmm, good point. Perhaps the physical universe then contains no circles at all and your formulas are but representations of an idealized universe. Useful represenations though, no? Yes, I see that if we posit that circles exist then so can this. I stand corrected, or perhaps enlightened is a better term.--Justanother 21:42, 25 August 2006 (UTC)

Now I'm really confused. I thought I was going to be able to tell if you were an engineer, or a mathematician. But your words imply you must be neither. The article, by the way, is about mathematics. Dicklyon 21:03, 25 August 2006 (UTC)

What does my profession have to do with the price of tea in China (yes, now I am an international economist). My point vis-a-vis the article is one of communication and logic, semantics if you prefer, not of mathematics. And I don't think I was picking a nit for the reasons given previously.--Justanother 21:45, 25 August 2006 (UTC)

Your comment re "difference in mindset between the mathematician and the engineer" led me to believe I was going to learn which mindset your were. Nothing at all about your profession. If you have a point about improving the article, please do re-state it, as it has long since been lost in the banter. Dicklyon 22:05, 25 August 2006 (UTC)

Two objections to this article

It seems to me that much of the arguements on this talk page revolve around the following, "The circle can be squared, but not with a compass and straitedge (or even with a ruler)".
It seems to me that this article essentially boils down to, "If we deny ourselves the tools that solve this problem (squaring the circle), then this problem is unsolvable. But, if we allow the use of the tools that solve this problem, then it is indeed solvable." It seems to me that such a statement could be made about most any complex mathematical problem. Although it was a tremendous mathematical breakthrough to demonstrate that the circle indeed cannot be squared through the use of a compass and straightedge and although it may be an interesting historical note to point out how many countless untold man-hours of work have been wasted on attempting to prove this one way or another, now that we can actually square the circle (albeit using tools other than compass and straightedge), this problem loses much of its meaning.
Furthermore, we have two statements which, when analyzed, would seem to create a logical fallacy.

• Since we can never exactly determine the precise value of the square root of pi, let alone the square root of pi, we can never truly draw a square with the same area as a circle.
• We can never truly draw a square with the area of a circle exactly, since pen/paper or computer pixels, whatever, can never be absolutely precise enough.

Since pen/paper, whatever, is inherently "not good enough", then can't we say that this problem is solvable, as pen/paper or computer pixels or whatever are able to come as close as we can calculate? In other words, even though the calculations are "off", so is the medium that we are using to represent the problem and the medium that we are using to represent the problem is off by a greater amount than our calculations are off by. Thus, we can draw a square with the same area as a circle, or at least it's the same area as far as we can accurately measure.
Note, there were two objections to this page in the preceding statements.

• If we deny ourselves the tools that solve this problem (squaring the circle), then it is unsolvable. But, if we allow the use of the tools that solve this problem, then it is indeed solvable.
• Although we can mathematically prove that we can never accurately draw a square with the same area as a circle, our drawing methods are "off" by a greater amount than our calculations are and thus, as far as we can determine based on our drawing methods, it can be drawn. Banaticus 06:19, 19 September 2006 (UTC)

Banaticus 06:19, 19 September 2006 (UTC)

But the arguments on the talk page are just that; arguments. Is there anything about the article that you see needs improving? Dicklyon 13:29, 19 September 2006 (UTC)

Banaticus' objections are silly. His first bullet point is right, but it's silly. The point is that the fact that those particular tools are inadequate is very very far from trivial. Yet Lindemann proved it, by building on the work of many predecessors. As far as "drawing methods" go, Banaticus' statement is obviously correct, even without any of the work of Lindemann or his predecessors. But who cares? It's really not relevant to this article. Michael Hardy 16:13, 19 September 2006 (UTC)

now that we can actually square the circle (albeit using tools other than compass and straightedge), this problem loses much of its meaning.

Again: silly! "Now that"?? As if we couldn't do that before? People have always been able to "square the circle" that way, but that doesn't deprive the problem of any of its meaning. The problem is not about drawing pictures. Everyone's always been able to draw sufficiently accurate pictures. Michael Hardy 17:28, 19 September 2006 (UTC)

Agreed! Really any considerations about actual drawings are totally irrelevant to this topic. A good way to think of this particular topic is to consider some of the highly complex problems that CAN be solved using ruler and compass (e.g. regular 17-gon, and many others), and compare these with the fact that squaring the circle does not belong to the class of "problems solvable by ruler and compass". Madmath789 17:37, 19 September 2006 (UTC)

"If we deny ourselves the tools that solve this problem (squaring the circle), then this problem is unsolvable. But, if we allow the use of the tools that solve this problem, then it is indeed solvable." I think a lot more emphasis should be placed in this article on the fact that this problem is, indeed, solvable. It's just that it's not solvable with finite methods. For instance, the opening image and paragraphs could be rewritten as follows:

Image -- Squaring the circle: the areas of this square and this circle are equal. In 1882, it was proved that this figure cannot be constructed using only finite methods, although it is constructible using other methods.

Squaring the circle is the problem proposed by ancient geometers of using finite methods (construction using an idealized compass and straightedge) to make a square with the same area as a given circle. In 1882, the problem was proven to be impossible using finite methods, although the problem can be solved using other methods. The term quadrature of the circle is synonymous.

Statements in the body of the article such as, "If one solves the problem of the quadrature of the circle, this means one has also found an algebraic value of π, which is impossible." And, "The mathematical proof that the quadrature of the circle is impossible has not proved to be a hindrance to the many people who have invested years in this problem anyway." These statements, taken at face value, imply that no method has been yet found for squaring the circle, that we cannot, using any method, square the circle. These statements (and others like them) should be rewritten so that the focus of the article is in the right place -- that squaring the circle is only impossible using finite methods.

Furthermore, perhaps there should be a greater emphasis placed on "finite methods" and defining exactly what that is. The article does already say, "It is possible to construct a square with an area arbitrarily close to that of a given circle. If a rational number is used as an approximation of π..." Yet, as I pointed out, this does mean that the circle can be squared using only the physical tools of compass and straightedge (as they differ from the nonexistant idealized forms of a compass and straightedge). Perhaps, in addition to placing greater emphasis in this article on the fact that this "problem" is solely a "well, let's limit ourselves to using only these tools then try to solve it" type of problem (which should greatly cut down on the number of people who say, "But wait, this problem has been solved!"), there should be greater emphasis placed on what finite methods are and how "finite methods" in abstract and an idealized compass and straightedge differ from the physical real life tools of an actual compass and straightedge. Banaticus 18:29, 19 September 2006 (UTC)

I'm going to look at the article within a few days and maybe alter the emphasis in spots, and add some further explication. Michael Hardy 23:08, 19 September 2006 (UTC)

More emphasis has now been placed in the article on the impossibility of squaring the circle only relating to a restriction that only finite methods can be used. The article still needs more work, though. Banaticus 19:08, 25 September 2006 (UTC)

Nice revision, Michael Hardy. :) Banaticus 20:59, 25 September 2006 (UTC)

Completely agree with Michael Hardy here: Banaticus's point is ridiculous. Not understanding what the phrase "Squaring the Circle" means he conflates it with another problem of drawing a near-enough accurate picture, or on the other side with the abstract existence of said square. Neither of these are at issue: at issue is solvability using particular methods connected to polynomial equations. Eluard (talk) 10:36, 15 March 2011 (UTC)

Michael Feldman's Whad'Ya Know on-line quiz

This week's question for the Whad'ya Know on-line quiz is "Can you square the circle?" The official answer is no, citing this article as the source. However, this article answers the question, "can you make a square of the same area as a given circle with only a ruler and a compass?" As numerous participants have noted above, without the straightedge-and-compass limitation, the circle can be squared. Just make a square with a side the length of the circle's radius times the square root of pi. I am disappointed to see descriptions of this problem that do not explain the straightedge-and-compass limitation. The quiz question is available this week at: http://notmuch.com/Quiz/ In later weeks, it will be available at: http://notmuch.com/Quiz/past-weeks.html r3 13:55, 30 October 2006 (UTC)

Now accessible at this link as of July 2009 Joe Hepperle (talk) 22:16, 6 July 2009 (UTC)

Squaring the circle and the longitude problem

According to De Morgan's A Budget of Paradoxes, there was a good deal of confusion in 18th and 19th century England on the problem of squaring the circle. Many believed that Parliament had established a prize for solving the problem. De Morgan claims that this was because people confused the problem with the Longitude Problem (for which a prize existed). Should this be added to the article? Magidin 15:54, 30 October 2006 (UTC)

I would think that would be worth a mention, if you have a good source. Dicklyon 16:01, 30 October 2006 (UTC)

Well, De Morgan makes the point several times; I can look it up and give precise quotes. He was often attacked (especially by a two particular individuals, Sir Richard Phillips and a Mr. Smith) of trying to "cheat them" our of their alleged parliamentary prize for their efforts at squaring the circle. He also connects the confusion with the Longitude problem. I'll look it up (my copies of De Morgan are at home). Magidin 16:37, 30 October 2006 (UTC)

Sorry for the long delay. I had other demands on my time, and it wasn't as easy to find as I thought. Here is was de Morgan says in A budget of paradoxes, pp. 96:

Montucla says, speaking of France, that he finds three notions prevalent among cyclometers: 1. That there is a large reward offered for success; 2. That the longitude problem depends on that success; 3. That the solution is the great end and object of geometry. The same three notions are equally prevalent among the same class in England. No reward has ever been offered by the government of either country. The longitude problem in no way depends upon perfect solution; existing approximations are sufficcient to a point of accuracy far beyond what can be wanted. And geometery, content with what exists, has long passed on to other matters. Sometimes a cyclometer persuades a skipper who has made land in the wrong place that the astronomers are at fault, for using the wrong measure of the circle; and the skipper thinks it is a very comfortable solution! And this is the utmost that the problem has to do with longitude.

Should I add a new section with some of this? Magidin 19:03, 19 November 2006 (UTC)

Sounds good to me. Dicklyon 19:14, 19 November 2006 (UTC)

For the particular supposed connection with the longitude problem I think an entire section would be too much. However, it might be feasible to split out the first paragram of the "as a metaphor" section into a brief discussion of circle-squaring as a favorite crank pastime, and your reference would fit perfectly there. The problem here is more that it is hard to write such a section in a way that will not attract POVness criticism. Henning Makholm 19:19, 19 November 2006 (UTC)

Sorry; I finished writing a short section on it before you posted your comment. I am not sure how it will attract POV criticism: it is certainly the case that 18th and 19th century circle squarers seem to have believed a prize existed and that it was connected with the longitude problem, and it is also a fact that no such prize was ever offered. I think it does not fit well within "as a metaphor", because this is not really part of using "squaring the circle" as a metaphor. On the other hand, I can add to that section the fact that the expression "Descubriste la cuadratura del círculo" ("you discovered the quadrature of the circle") is a common expression in Mexico, as a derisive response to someone who claims to have found the answer to a particularly difficult problem... Magidin 19:45, 19 November 2006 (UTC)

My point was that the first paragraph does of "as a metaphor" is not about metaphors eihter, and this and your quote together might make a viable new section about crank circle-squarers. (I'd do this if only I could think of a good section title). Henning Makholm 19:50, 19 November 2006 (UTC)

Ah. Okay, I moved that paragraph and retitled the second section to "Claims of circle-squaring, and the longitude problem". Maybe another title might be better, but "claims of" seems to me to be neutral POV. Magidin 20:08, 19 November 2006 (UTC)

Looks good. Henning Makholm 20:14, 19 November 2006 (UTC)

Constructible numbers

This recent anonymously edited paragaph was reverted:

In 1882, the problem was proven to be impossible, as a consequence of the fact that that pi (π) is transcendental, not algebraic; that is, it is not the root of any polynomial with rational coefficients. That the transcendance of π would have that consequence had already been known for some decades; but the transcendance was finally proved in 1882. Since π is not algebraic, it is also not a member of the subset of algebraic numbers which are constructable (which only include algebraic extensions of the rationals that are compounded from a finite series of solutions to quadratic equations). Once it was proved that constructable numbers were members of such extensions, it was easy to prove that the circle could not be squared (as π is not even algebraic), 60 degrees could not be trisected (it requires solving an irreducible cubic) and the cube could not be doubled (it requires constructing the cube root of 2). Approximate squaring to any given nonzero tolerance, on the other hand, is possible in a finite number of steps, corresponding to the fact that there are members of the specific field extensions of the rationals arbitrarily close to π.

...with comment "Revert addition which I think has it backwards: The major features for constructible numbers (including that they are all algebraic) had been known for at least decades before 1882."

Now, I don't disagree with it being reverted, but I do disagree with the reason. It is not incorrect, nor is the fact that constructable numbers were long known at variance with what it says here was proved in 1882. But this background on constructable numbers is not necessary to support the main point, which is that pi being not algebraic proves it can't be constructed. And it's too much side detail for the lead section. Put it into the history section instead, preferably with something about when it was shown. Dicklyon 02:16, 2 November 2006 (UTC)

Let me explain what I think was wrong. We consider the three propositions:

• A: The constructible numbers form a certain algebraic extension of Q.
• B: Pi is trancendental.
• C: Squaring the circle is impossible with Euclidean tools.

As far as I can read the anon's text, it says: "Once A was proved, it was easy to prove C (because B holds)". We all agree, I hope that C indeed follows from A and B, but I disagree with the implication that A was the last missing bit in the proof of C. My understanding of the historical development is the opposite: A had been known for decades before B was proved, and then, "once B was proved, C followed immediately (because it was well known that A holds)". Am I making my point clear? Henning Makholm 19:45, 2 November 2006 (UTC)

Yes and no. I understand what you are saying, but I don't understand why you read it that way. B was clearly the last missing bit. It says B was found out: "In 1882, the problem was proven to be impossible, as a consequence of the fact that that pi (π) is transcendental," and it says that it had been known that if that could be shown, then game over. Your "A" is more narrowly drawn, but knowing that pi is transcendental was well known to be enough to know what it was not constructible. Perhaps the way it is worded confused you. It could perhaps be more clear. But no need to introduce the more narrow definition of what is constructible here, as long as it's clear that transcendentals are not. Dicklyon 05:46, 3 November 2006 (UTC)

The sentence "In 1882, the problem was proven to be impossible, as a consequence of the fact that that pi (π) is transcendental" is not the one I think it wrong. I am speaking about the sentence "Once it was proved that constructable numbers were members of such extensions, it was easy to prove that the circle could not be squared (as π is not even algebraic)". I have trouble understanding why you don't think this sentence declares A to be the last missing bit - otherwise, what is the point of the word "once"? Henning Makholm 19:35, 3 November 2006 (UTC)

Oh, I see. We are having a violent agreement. I thought you were quibbling about the current state of the article, but you're talking about the part some guy put in and you took out. Sorry I got out of sync; too much output, not enough input. On re-reading what you reverted, and your reason, I guess I do pretty much agree with you on that. Sorry for the confusion. It might still be worth representing that idea of constuctible numbers, and the history of it, in the article, but not the way it was in the lead. Dicklyon 20:17, 3 November 2006 (UTC)

Good. For future reference (and because I have now bothered to look it up): According to V.J. Katz A History of Mathematics (HarperCollins 1993, pp. 597f), the algebraic properties of the constructible numbers were investigated by Gauss in Disquisitiones Arithmeticae, and their algebraic characterization was completed by by Pierre Wantzel in 1837, which closed the angle trisection and cube doubling problems. Henning Makholm 23:38, 3 November 2006 (UTC)

You're all wrong...

It appears that not a single one of you remembers the Definition for Numbers: A NUMBER IS AN IDEEA! I shall add ... A circle is made of points which are also ideeas of locations. A line and a curve are made of points . This are idealized concepts which only exists in Thought. The manifestation of this ideeas in the "physical univers" are good enough for the purpose of visualization only. Pi , sqrt 2 , and the rest of irrational numbers have geometric form ( a triangle 1,1,rt2, or a circle with diameter 1- circumference = pi ) . Squaring the circle is just ... "matter for thought." Just like Zeno's Paradox ... we can't ever draw anything completely ... therefore PI . All is just an illussion! ("deceptive appearance" ) —Preceding unsigned comment added by 75.50.147.71 (talk) 2007-04-03T14:52:04

(The transcendence of _ implies the impossibility of exactly "circling the square", as well as of “squaring the circle”). This is true as long as we look at it in Two Dimensional view. But if we look at it in Three Dimensional view, we will have a Cube and Sphere. The Cube Ratio: Cube Height : Cube Square Diagonal : Cube Rectangle Diagonal = √1 : √2 : √3 The Sphere Ratio: Sphere Diameter : Sphere Circumference = √1 : (√2+√3) In this perspective the Pi Ratio is solved: π = (√2+√3) = 3.14626436994197 Hanaga 16:32, 9 June 2007 (UTC)

Nope. Any combination of √n is algebraic; π is transcendental, i.e. not algebraic. —Tamfang (talk) 14:31, 12 July 2009 (UTC)

Diagram could use fixing

Image:Hipocrat arcs.svg has a freaky aspect ratio such that the "circles" are pretty obviously ellipses, which sort of works to disorient the reader. To some extent, of course, it won't be possible to get perfect circles on all screens, but I expect we can get far closer than the image currently does. Also, I apologize for not just fixing it myself-- I have a phobia of vector graphics. --Alecmconroy (talk) 15:32, 27 January 2008 (UTC)

Article

The Wikipedia article on the Hobbes-Wallis controversy could be mentioned. —Preceding unsigned comment added by 87.194.34.71 (talk) 13:28, 21 April 2009 (UTC) Hobbes himself is mentioned, but not the separate article. —Preceding unsigned comment added by 87.194.34.71 (talk) 11:54, 22 April 2009 (UTC)

Additionally, this article is slightly misleading. While Hobbes did describe himself as having "squared the circle", it is more accurate to say that, on the assumption of a certain value of pi that Hobbes himself admitted was an estimation, he showed a geometrical construction (line and compass) of a circle with a same area as a given square. — Preceding unsigned comment added by 50.73.127.69 (talk) 19:54, 4 August 2011 (UTC)

Squaring of the Circle, Wikkepedia

We find it discouraging that Wikkepedia can be very dogmatics and mathematics can be dogmatic about such issues although Wikkepedia is as good as its feeders . We have clearly proven that Wikkepedia is wrong when it says that "circle is not squarable" . We have proved it wrong, and we will below enter our clear method and proof and also reference this new mathematics that has the diagram of a square of a circle at the end of the diagram section using a brand new mathematics. ----www.inverese19 mathematics .com .. please punch and rview . Our complete method will be posted below —Preceding unsigned comment added by Dodged (talk • contribs) 17:37, 25 July 2009 (UTC)

Freemasons

Is "squaring the circle" (using a somewhat a posteriori approximation to π based on the golden ratio) part of initiation into freemasonry? 74.98.43.217 (talk) 22:08, 1 October 2009 (UTC)

claims of squaring the cicle and new value of PI

There are clams that PI is exacly 3.1416 and that you can square the cicle by means of Ruler and Compass. http://www.alkyone.com/mak-pi-gr/en/en_release.htm

The person who makes this claims is Mr. Moschos Ath. Karagounis, Mechanical Engineer of the National Technical University of Athens (NTUA).

There is quite a talk about this recently. Now, at first i thought this to be pseudomathimatics but the fact that this person is part of greek academia and my maths are not to his level, makes it imposible for me to disprove it. So, I write this post, so a more advanced wiki editor could make sence of it and decide what to do with it.

Also, There is another page, http://www.squaringofthecircle.gr/ where he gives another number for Pi , but with somewhat same claims.

Nkast (talk) 10:13, 10 October 2010 (UTC)

See Ferdinand von Lindemann. People can say whatever they like in self-published mathematics papers on the web, but Pi is considered to be a transcendental number. See also WP:REDFLAG.--♦IanMacM♦ ^{(talk to me)} 10:18, 10 October 2010 (UTC)

Archimedes Solution

I have a problem with 'Archimedes Solution' section of the article. The line tangent to point P on the circle can never intersect the axis at point T. By definition, the line tangent to the circle at point P is parallel to the indicated axis. Is a better citation warranted? — Preceding unsigned comment added by 173.65.133.176 (talk) 13:29, 1 June 2011 (UTC)

You've misread the part of the article. P is not in the circle; as the article says, "Let P be the point on the spiral when it has completed one turn." The tangent in question is tangent to the spiral at point P, not to the circle. Magidin (talk) 17:03, 1 June 2011 (UTC)

I apologize that I am years out of school, and a lot of this knowledge has left me. How does Archimedes solution (creating a triangle with the area of a circle) solve the issue? Is there a method of forming a square from any triangle, using a compass and a straight edge? If so, we should add the relevant link/explanation. --Bertrc (talk) 00:40, 7 August 2011 (UTC)

Additionally, instead of merely referencing his first proposition as though it was some compllicated algorithm, can't we just say that the area of a circle (pi*(r^2)) equals the circumference of the circle (2*pi*r) times the radius of the circle (r) divided by 2, while the area of all right triangles equals the long leg (in this case, the circumference of the circle) times the short leg (in this case, the the radius of the circle) divided by 2. This is clunky when written out, but for a wikipedian who is good with formulas, it could be displayed rather concisely. -- Bertrc (talk) 00:26, 7 August 2011 (UTC)

Lastly, if a square can be made from any right triangle, is there a proof that Archimedes spiral cannot be constructed with a straight edge and compass, other than the fact that doing so would allow the squaring of a circle? -- Bertrc (talk) 01:05, 7 August 2011 (UTC)

Other Tools

So squaring a circle with just a straight edge and compass is impossible. Has anybody ever thought of or written about other tools that can solve this? eg, I could wrap the circumference of a circle with a length of rope (or with a piece of paper turned perpendicular to the plane of the circle, or I could cut out the circle and roll it, etc.) If I straighten the rope (or paper), I now have a line of length 2*pi*r. if r is 1, I can easily split the line in half to get a line of length pi. Given a line of length x, is it possible to construct a line of length sqrt(x) (with which I could easily construct a square) or is that some other unsolvable problem? That was just an idea off the top of my head. Are there other tools that people have considered (like folding the paper, etc.) --Bertrc (talk) 01:02, 7 August 2011 (UTC)

See quadratrix; Archimedean spiral. —Tamfang (talk) 08:30, 26 February 2013 (UTC)

Good question. I was wondering about that too. All that's mentioned is construction and algebra in Euclidean space. I want to know if other methods have been tried, like trying to express pi in sin/cos/tan functions or maybe in a non-base-10 number system. If anyone tried and failed at such methods it would be nice to mention it in the article.

Constant: "Square Root of Pi" and "Half of Pi"

I apologize -- I am not skilled in Wikipedia.

There will always be conflict when we do not differentiate between "concepts" and "representations of concepts". Any form generated by hand is purely a representation. Pi is misused when we attempt to represent it in some partial form, and then cry foul when it is appreciated in conceptual form. As you can see from the link below, the areas of a circle and square can be exactly the same (even in conceptual form).

What seems important is to acknowledge that there is a constant that exists: "Simply put, the pure geometric difference between a circle and a square lies in the difference between 'the square root of pi' and 'half of pi'" - Thomas Bishop

Please refer to this link:

http://c.ymcdn.com/sites/www.optimalmindsgroup.com/resource/resmgr/Divosophy/BishopSquareCircle.png

Bishopclinics (talk) 05:49, 6 January 2013 (UTC)

Of course there exists a square with the same area as any given circle. That doesn't have anything to do with the problem of squaring the circle, which is to construct a square with the same area as any given circle using only compasses and straightedge. The compasses and straightedge aren't actual compasses and straightedge, rather they are idealised compasses and straightedge with properties not found in real objects (e.g. the straightedge is infinitely long). Hut 8.5 11:33, 6 January 2013 (UT

Related Areas

When you superimpose a circle over an equivalent area square, as in the illustration, you note that there are 4 related areas that are outside the circle but inside the squire and also also 4 other related areas that are outside the square but inside the circle. Then a little logic will tell you that the individual areas of all of these related areas is the same constant value. So what I would like to know is what the precise value of this constant is, and does it have any mathematical relationship with the area values of any of the other sub areas of either the circle or the square. I think its value is approximately 7.5% of the area of the square (and also the circle) But I would like to know its value to a precision equal to that of the other sub areas. Can I find that information somewhere?WFPM (talk) 01:12, 11 May 2013 (UTC)

New Comment

I give a compass & straightedge construction using the Golden Mean which yields 3.14164 for pi at http://www.goldennumber.net/squaring-the-circle/ Ricci4.4428828 (talk) 00:58, 11 June 2013 (UTC) I have also discovered a compass & straightedge method of squaring the circle precisely in both area & perimeter on smooth Riemannian manifolds of positive Ricci curvature; despite the transcendental nature of pi: http://www.circleissquared.com/index.html C. Ricci Ricci4.4428828Ricci4.4428828 (talk) 00:58, 11 June 2013 (UTC) — Preceding unsigned comment added by Ricci4.4428828 (talk • contribs)

See what Wikipedia is not. If you are proposing additions to the article, then what you point to above cannot be included: it violates the no original research policy, there are problems with its notability, and you have a conflict of interest. And if the above is simply meant to be calling attention to your work, then it is also inappropriate for a talk page; talk pages are for discussions on improving the main article, not for discussions on the topic of the main article. That is, talk pages are not an internet forum. If your addition above is not meant to be either, then please explain what it is meant to be. Magidin (talk) 03:25, 11 June 2013 (UTC)

Also see Ferdinand von Lindemann, which points out "Before the publication of Lindemann's proof, it was known that if π was transcendental, then it would be impossible to square the circle by compass and straightedge." My calculator says that the value of π is 3.1415926, not 3.14164.--♦IanMacM♦ ^{(talk to me)} 05:11, 11 June 2013 (UTC)

This value for pi (3.1416)is derived from the Fibonacci numbers; a sequence of continued fractions which ultimately converge on the Golden Mean. I give a compass & straightedge construction using this ratio which yields the same value for pi at http://www.goldennumber.net/squaring-the-circle/ These results have been verified by the website host Gary Meisner, certified CPA and an MBA in Finance. Ricci4.4428828 (talk) 11:52, 11 June 2013 (UTC)

It is worthy to note that this tight approximation for pi (3.1416) using the Golden Mean is mentioned on many math related sites. For example: http://mathworld.wolfram.com/PiApproximations.html. I am not sure if I referenced that correctly. Ricci4.4428828 (talk) 12:21, 11 June 2013 (UTC)

The classical problem of squaring the circle requkres the construction of a square that has the exact are as that of given circle. The exact area; so any approximation of pi, no matter how it is derived, is irrelevant. The construction must give the exact area, which requires the use of pi, not any approximation thereof. And "verifications" by a CPA/MBA are irrelevant. That's not a recognized reliable source. See Wikipedia's policies on reliable source for information on what is and what is not considered a reliable source. "[W]ebsite host Gary Meisner, certified CPA and an MBA in Finance" is not one. Magidin (talk) 15:42, 11 June 2013 (UTC)

I think that 2 separate considerations are being confused here. The exact value of Pi has no meaning in itself. What is important is its value as a measurement of the relationship of 2 significant physical dimension values. If I create a square with an area dimension of 4, I can then say that the area within the enclosed circle is equal to the value Pi. And if I create a square with an area of 16, I can also say that the area within a 1/4th quadrant of the circle also has an area of Pi. So we are dealing with relative area relationships here, of which Pi is only 1. So these 2 different configurations of enclosed areas have the same relative size relationship, and the value of Pi is involved in the geometry of construction of the considered areas, and I have my doubts about the ability of a mathematical formula to come up with an exact numerical number for the value of Pi.WFPM (talk) 17:35, 14 June 2013 (UTC)

Pi has plenty of applications as a dimension-free number in mathematics. See Basel problem and Stirling's approximation for two of the most basic and important of them. My opinion is: if you have vague feelings that mathematics doesn't work, and you think it's more important to express your feelings than to understand clearly and communicate to others the consensus of published mathematics, then perhaps other topics than mathematics would be a better fit for your Wikipedia editing. —David Eppstein (talk) 17:56, 14 June 2013 (UTC)

If we roll a 1 inch diameter circle along a number line starting with a point at zero, the point will next touch the number line at the value of Pi. So it's our lack of the proper drawing tool that prevents us from drawing a straight line with a value of Pi. And I'm really not interested in exact mathematical values, and particularly in base 10 (or other base) values. Because I believe that all numerical values are merely names of relative magnitude locations along a number line and accordingly of no significance unless related to some physical article of consideration. However, I don't think that I should be disqualified for pointing out that there are other fixed relationships of areas of the circle versus that of the equal area square that might offer an idea as to the logic of those relationships.WFPM (talk) 03:39, 15 June 2013 (UTC) PS: I looked at your Basel problem reference and wish to compliment Euler for getting the value Pi squared out of a summation process related to a mathematical formula.WFPM (talk) 04:00, 15 June 2013 (UTC)

You may not be disqualified from pointing out things in many places. However, insofar as you wish to point out what you term to be your "beliefs" and your "interests" (or lack thereof), they have no place in Wikipedia as they constitute original research and are both unverifiable (within the meaning of Wikipedia) and lack notability (again, within the meaning of Wikipedia). The comments are also out of place in talk pages, as talk pages are not fora for discussion, but rather places to discuss improvements to the articles, improvements that are supposed to be based on information that is verifiable, from reliable sources, and notable. As such, I must echo David Eppstein comments: your contributions here are out of place. Magidin (talk) 19:37, 15 June 2013 (UTC)

My Gosh!! We're talking about the square area that is equivalent to that of a circle and now it's morfed into a discussion about the value of Pi. I'm interested in the area of a square and the equal area circle and I came to here for information. And now you don't like my comments about the significance about the importance of the value of Pi. And where else on Wikipedia can I go for this information? You don't care! And you say the value of Pi is 3.1415926 and my 8 dollar Casio calculator says it's 3.141592654 so what?. What I'm interested in is that if I consider a square with area 4 then draw a circular 90 degree arc through that square which has a length of Pi and separates the area of the square into the 2 sub areas Pi and 4 - Pi. So how does that fact interact with the fact that the areas outside of the circle or the equivalent area square are equal in area? Aren't these ideas relevant to the subject matter of the squaring of the circle.WFPM (talk) 23:02, 15 June 2013 (UTC)

I don't know what it is you think you are talking about; for example, I never said the value of Pi "is 3.1415926". Nor did I say anything about my likes or dislikes, I said your comments are out of place in this talk page; and, you know what? They still are. Your interests are likewise immaterial, and your continued discussion of them here is out of place. No, your ideas are not relevant to the improvement of this article, because they continue to be original research, unverifiable, and lack notability. And, for that matter, betray willful ignorance on your part. Magidin (talk) 01:57, 16 June 2013 (UTC)

Well Okay! I'm sorry I bothered you about any detailed information about the geometric properties of the equal area square with relation to the circle. I thought it might be relevant to the subject matter of this article.WFPM (talk) 03:29, 16 June 2013 (UTC)

The illustration on the left is wrong

A circle with a radius of one has an area of pi squared. A square of equivalent area has sides of length pi, not the square root of pi as the illustration shows. Kriegshauser@iit.edu (talk) 19:17, 13 January 2015 (UTC)

No. The area of a circle is $\pi r^{2}$, where $r$ is the radius. For $r=1$ the area is $\pi 1^{2}=\pi$. Hut 8.5 19:38, 13 January 2015 (UTC)

Huh?

Like an increasing number of scientific or technical articles on Wikipedia, this seems to have been written by experts to trigger discussions between their colleagues and would-be experts. Meanwhile, many people who want to know what "squaring the circle" actually means have to look elsewhere. "...the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge." means what exactly? We can't measure the area of the circle? When "constructing" squares (from wood?) it's tricky getting the edges straight? And why would you use compasses to "construct" a square? An article this obscure is worthless, unless it helps a math student who was assigned to write it to get a good grade.

To understand the that sentence see Compass-and-straightedge construction. Paul August ☎ 16:26, 7 March 2015 (UTC)

Circularity?

I read the discussion above because I found the article unclear about one step -- not the one discussed so much above, but rather: *Why* it is impossible to "construct" an irrational value from a rational one? I make my living doing other sorts of applied math, but had not encountered the formal term "a constructible number" (minimal extension of the field *R* such that...). Given a typical reader's likely intuitive idea of "constructible", the question arises: "Why doesn't setting a compass to some given size (say, 1), and then merely using it to draw a circle, amount to "constructing" a transcendental number (2pi) from a non-transcendental (1)?" Doing so obviously "constructs" a locus of (virtual graphite) points whose length is a transcendental multiple of 1 -- but the intuitive meaning of "construct with compass and straightedge" differs subtly (but crucially) from the formal definition you need here. When you try to figure it out by reading constructible numbers, it's clear that the notion "constructible" originates from "constructible with compass and straightedge" (it does give a field-oriented definition as well, but the equivalence is not trivial/obvious). So the current explanation boils down to "You can't square a circle with compass and straightedge, because pi is not a constructible number. What's a constructible number? One that can't be constructed with compass and straightedge." That seems to beg for a fix. — Preceding unsigned comment added by Sderose (talk • contribs) 16:35, 29 October 2015 (UTC)

After selecting an arbitrary length to be considered the unit (value 1), we say that a positive number a is constructible if we can mark two points p and q on a straight line, using only compass and straightedge and the given unit value, such that the distance between p and q is exactly a times the unit. The fact that the length of the circle is 2pi does not actually yield a construction of a straight line segment of length 2pi. The actual explanation does not, however, boil down to what you claim it does. The actual argument is: We can square the circle if and only if we can construct the number ${\sqrt {\pi }}$. If we could construct that number, then we would also be able to construct $\pi$. However, $\pi$ is transcendental, and no transcendental number is constructible. Since $\pi$ is not constructible, neither is its square root, and since its square root is not constructible, we cannot square the circle. The argument does not rest on "pi is not constructible", it rests on "pi is transcendental." Magidin (talk) 17:18, 29 October 2015 (UTC)