Talk:Squaring the circle

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[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

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