Being Square

I was just thinking about the expression "square the circle", and I realize I'm not totally sure about what it means. People were trying to construct a square with an area identical to a given circle...using only a compass and a straight-edge, maybe? Must be, because otherwise...hm. But you really can do so in non-Euclidean geometry, IIRC? But can your straight-edge be getting a lot of play in the non-Euclidean world, or do you have an infinite series of curved-edges to use? That could take up a lot of room. Someone help me out here.

This just reminds me that one of the things I have looked forward to in having school-age children is that I will very slowly get to re-learn all of math, at a painless pace. All the worries I'll have when Zoë and Violet are teenagers will surely be compensated by the fun of re-learning the calculus, right? That was the only part of HS math I liked. That, and graphing imaginary numbers on those wonky polar graph things.

John has sucessfully explained zero to Zoë. She was counting the people in the elevator as they left, until only she and I were on, and when we got out, she said "and now there's zero people in the elevator!" Right on for the null set, baby! I ran into a little trouble trying to explain infinity the other day, however. All in good time, I guess.

Comments

[ben wolfson]

Mathworld to the rescue! "While the circle cannot be squared in Euclidean space, it can in Gauss-Bolyai-Lobachevsky Space."

[Walt Pohl]

Yes, "squaring the circle" is restricted to compass and straight-edge. Squaring the circle in that case is impossible because you need to construct a line segment of length square root of pi, and you can only construct lengths of iterated square roots of integers.

In non-Euclidean geometry, you would use the "straight lines" of that geometry, which (if you are representing the geometry using a curved surface as a model) would be geodesics -- paths that minimize the distance travelled.

The result about squaring circles on the hyperbolic plane is new to me, though. Is it well-known?

[dsquared]

You can avoid this problem by doing what I'm doing and teaching your children mathematics in the logical (ie set-theoretic) order. My three-year-old has just mastered the concept of an equivalance relation and now we're moving onto homologies. By the time he's eighteen, he will finally be ready for 1+1=2 and he will understand it.

[des von bladet]

Set theory? In the 21st century? That's practically child abuse; we all use categories now.

[Standpipe Bridgeplate]

What Des said. Set theory's status as the preferred foundation is an artifact of path-dependence. Free your mind – reason up to isomorphism.

[Jeremy Osner]

There's a good explanation of infinity in "The Phantom Tollbooth", which Zoe is probly about ready for. (I read it to Sylvia a few months ago and she loved it, though without getting everything.)

[rob helpychalk]

The set theory approach is also borderline child abuse because it confuses a logical order with an epistemological order. Like Aristotle says (in the Metaphysics?) all inquiry begins with what is most known and ends with what is most knowable. The foundations of mathematics may be what is most knowable, but what is most known are small collections of plastic toys that you can do arithmetic with.

There was a period of time when educators were genuinely trying to use set theory in elementary school. It didn't work.

[sr]

I taught my daughter that "infinity" just means "...", ie indefinite continuation, an idea I got from LW. With this definition, questions about infinity being a number don't arise. But N. may not be a typical daughter . . .

[sr]

Also, I don't really understand the sentiment behind the judgement of the impossibility of squaring the circle. After all, there is surely a procedure using only straightedge and compass that iteratively approximates the correct solution, rapidly converging on the "correct" answer. Why doesn't this count as squaring the circle? Because there is always an error? But there is also an error in drawing a straight edge, or using a compass. Oh, but we are concerned with an ideal straightedge and an ideal compass? Why not an ideal iterative procedure?

[ben wolfson]

Because life is polynomially bounded.

[Walt Pohl]

sr: Because that's not the problem. The problem of "squaring the circle" means squaring the circle exactly.

[Walt Pohl]

The attempt to rewrite the foundations of mathematics in terms of category theory is evil and wrong. It is the fourth great evil we have been called to face down: after Nazism, Communism, and Islamo-fascism, it is our destiny to confront Catego-fascism. I'm not surprised to see dsquared on the side of the catego-fascists.

[LizardBreath]

But sr is right, an ideal (infinitely extended) iterative process gives you just as exact an answer as an ideal conventional construction.

I think the only answer to sr's query is that the ancient Greeks set the rules, and they wouldn't have bought the idea of an ideal iterative process.

[s]

Thank you for the moral support, o exhalation of the reptile. On teaching little children abstract algebra (the "new math"): I know a lady mathematician who says that she loved this approach as a child. Apparently, if one is destined for mathematics, it isn't a bad approach to pedagogy. For others, however, --

[bza]

To Belle, as a follow up on Walt's explanation of what a straightedge would be in non-Euclidean space:

What it means to say that a space is non-Euclidean is, roughly, that lengths and angles don't combine in the way they do in Euclidean space. Example: On a plane, if you walk north one unit and east one unit, you end up in the same place as you do when you walk east one unit and then north one unit. Now imagine you're on a sphere. If you walk one unit forward, turn ninety degress to your right, and walk another unit, you don't in general end up in the same spot as you do when you first turn to your right, walk one unit, then turn left and walk one unit. Movements that are equivalent in one space are not equivalent in the other.

However, this doesn't mean that there isn't a notion of straightness that applies in a non-Euclidean space. When you're pacing off one of those one unit lengths on the sphere, you are, from your perspective on the sphere, always travelling in the same direction. So the straightedge you would be using is still a single tool, as it were: The tool for marking off non-deviating paths.

To Walt: Isn't the necessary condition for constructibility being an iterated square root of a rational, not an integer? (That is, a real is constructible only if it lies in a quadratic extension of a quadratic extension, etc., of the rationals. This isn't sufficient, obviously, since then we could trisect the angle, but on your characterization we couldn't bisect the angle.)

[bza]

Aristotle has another reamrk (in the Posterior Analytics, around 103b I think) that applies even more directly to the idea of teaching kids set theory before arithmetic, namely that priority in the nature of things is distinct from priorty in knowability, and it's a mistake not to accomodate ourselves to that.

[bza]

But sr is right, an ideal (infinitely extended) iterative process gives you just as exact an answer as an ideal conventional construction.

Not really. The way we make mathematical sense of a claim that an infinitely iterated process gives an exact answer is to read it as the claim that we can achieve any specified degree of precision by allowing the process to continue for a sufficiently large but still finite number of steps. There doesn't seem to me to be anything arbitrary at all in seeing the problem of finding that kind of construction as simply different from the problem of finding a contruction that gives an exact answer in a finite number of steps.

[sr]

There doesn't seem to me to be anything arbitrary at all in seeing the problem of finding that kind of construction as simply different from the problem of finding a contruction that gives an exact answer in a finite number of steps.

I didn't say that the distinction between the two kinds of ideal exactness was arbitrary, I said that I didn't understand the sentiment. Lizardbreath correctly pointed out that the Greeks "they wouldn't have bought the idea of an ideal iterative process".

However, think about pi. Do we really want to say that if I draw a circle of unit diameter, the circumference is "exactly" pi, but that pi defined as a converging series is "inexact"? Doesn't the finite action of drawing a circle itself sum an infinite number of actions? (Zeno)

It was an origami technique that inspired this line of thought. A length, or an angle, can be folded in half. Given this, there is an iterative folding method for dividing into n equal parts. Discuss.

[Walt Pohl]

bza: You're right. It's rationals, not integers.

sr: Ultimately, the reason is because for 2000 years exactly squaring the circle is the problem people cared about, and not approximately squaring the circle; all we can do is speculate why. I think the reason is that squaring the circle is impossible, but for most of the history of the problem it was not known definitively to be impossible, so it was a tempting challenge.

[dsquared]

It's even worse than that; I'm teaching my son mathematics on the basis of Zermelo's axiomatisation and my daughter on the basis of Aczel's. So when they grow up, they will make exactly the same mathematical judgements but will never be referring to exactly the same entities (demonic cackle).

[Carlos]

I think I'll just give my kid(s) a brightly-colored calculator and ask 'em to help out Mom and Dad. Worked for me.

[Matt Weiner]

Walt--I also think that the problems of squaring the circle and trisecting the angle, in their exact forms, are so well remembered in part because the techniques used to demonstrate their impossibility are so important. Galois theory has many other applications (as you know much better than I, I'm sure).

Didn't the Academic Francaise declare that it was going to stop looking at proposed squarings/trisections before impossibility had been proved? I'm trying to use this as evidence that the proof was more interesting than the problem, but it may not be (especially if it's not true).

[Walt Pohl]

Matt: I think you're right. In the context of modern mathematics, the proof of impossibility is more interesting than the problem itself, since it uses techniques that can be used to solve other problems.

[SusanC]

Isn't Euclid's proposition XII.2 an iterative procedure that approximates the area of the circle to any desired degree of precision?

But this isn't considered the same as finding an exact (and, implicitly, finite) construction.

[Anton Sherwood]

There's no such thing as a square in nonEuclidean space!

[raj]

I'm actually surprised that nobody has mentioned Riemann, the "inventor" as it were of modern geometry.

Squaring the circle in Euclidean geometry is virtually impossible because "pi," the metric used in determining the area of a circle, is a transcendental number. It's as simple as that. The value of pi has been approximated out to thousands of digits. One can approximately square a circle, but not exactly.

[engels]

dsquared - I believe Paul Benacerraf has already tried this, with rather unhappy results.

…To return in closing to our poor abandoned children, I think we must conclude that their education was badly mismanaged – not from the mathematical point of view, since we have concluded that there is no mathematically significant difference between what they were taught and what ordinary mortals know, but from the philosophical point of view. They think that numbers are really sets of sets while, if the truth be known, there are no such things as numbers; which is not to say that there are not at least two prime numbers between 15 and 20. (“What Numbers Could Not Be” 294)

[ogmb]

Squaring the circle in that case is impossible because you need to construct a line segment of length square root of pi, and you can only construct lengths of iterated square roots of integers.

But pi is 3!

[Anton Sherwood]

And what is more transcendent than the Trinity?

[air yeezy]

Theory is gray, evergreen tree of life!