Reading John and Belle’s blog, not the place I would usually look for unfamiliar maths results, I discovered that the circle can be squared in Gauss-Bolyai-Lobachevsky space . Checking a bit further, I found this was not a new result but was shown by Bolyai back in C19.
I haven’t found a link that shows how the construction was achieved, though. Can someone point me in the right direction, please?
John Quiggin 
GBL space is constant negative curvature, which is also called an “anti-sphere”. I am rusty, but isn’t it necessarily locally Euclidean?
If so, then that is a weird result because it would seem to imply that the number of steps required to square the circle must grow without bound as the size of the circle shrinks [it is impossible to square the circle in Euclidean space, or equivalently, any algorithm to do so must have an infinite number of steps. Therefore an algorithm to square the circle on a locally Euclidean surface must require an unbounded number steps as the size of the circle goes to zero, because the surface is approaching a Euclidean surface at the scale of the circle]. Or am I missing something?
If I only had a brain.
John: think hyperbolic geometry – or when space is saddle shaped. The picture involves should help you to visualize it. That’s not the answer you are looking for, but it’s at least one step in that direction.
Presumably it is still impossible to square the circle on a sphere (constant positive curvature). So what is it about the negative curvature that undoes the problem?
I think I can see how my first objection may be invalid. Since the curvature is constant and negative, the non-Euclidean perturbation to the circle’s area in one direction is always countered by the non-Euclidean perturbation in the orthogonal direction. This is true no matter how small the circle is – so maybe there is some kind of almost “topological” canceling that you can exploit on the negative-curvature surface that can’t be exploited on the positive curvature surface since the perturbations reinforce rather than cancel there.
Some Googling turns up this book with a description of the result.
It occurs to me that the construction might not be a general construction, but only a construction for particular circles. Also, you can’t yet jump to the conclusion that Bolyai actually found a construction; while he may well have done, it is also possible from that wording that he merely found an existence proof for a construction.
I think anon is overstating the problem. It would be possible to square the circle in Euclidean space if you had a construction for pi as a length relative to a preferred unit. If you have that, squaring the circle is simply a matter of determining the radius, a geometric square root and multiplication, and then drawing lines to scale.
I bet these guys can give a simple answer and perhaps a realtime graphic
http://www.cs.unm.edu/~joel/NonEuclid/NonEuclid.html
I now know how to draw a circle on trumpet, just not sure what the area is.
To Ex-Anon:
You say:
“[it is impossible to square the circle in Euclidean space, or equivalently, any algorithm to do so must have an infinite number of steps”
It is not obvious to me that these two statements are equivalent. Do you have a reason for saying they are?
I don’t think there’s an answer to your question, John.
The reason why you can’t square a circle in Euclidean geometry is that pi is transcendental – it’s not a solution to any polynomial equation with integer coefficients. The operations allowed in the “squaring the circle” problem (i.e., those possible with a compass and straightedge) can always be described by such a polynomial. Hence, no squaring of the circle.
In non-Euclidean geometry, the circumference:diameter ratio takes on values other than pi. If that ratio isn’t transcendental, then it’s possible to square the circle.
The series of operations required, though, depends on exactly what the ratio is. If the ratio was 2, you’d perform a certain set of operations. If it was 2.5, or 3, or 3.5, you’d perform completely different sets of operations.
For a simple example of how pi can vary, consider the geometry of the surface of the earth. Take the north pole as the centre of some concentric circle drawn on the earth – the lines of latitude. The radius of each circle is the distance from the pole to the line of latitude.
Very close to the north pole, the circumference:diameter ratio is approximately equal to pi. At the equator, the ratio must equal 2. (Assuming that the earth is perfectly spherical).
I haven’t read Bolyai’s paper, but as far as I know, what it does is establish the validity (non-contradictoryness?) of non-Euclidean geometries. Bolyai’s paper was published in 1831, and the proof of pi’s transcendentality wasn’t published until 1882. I do not find it credible that Bolyai would’ve attempted a proof of cricle-squaring in such a paper, when the proof the circle-squaring in Euclidean geometry was impossible lay some 50 years in the future.
That should read: “For a simple example of how the circumference:diameter ratio can vary,”.
SJ, I was surprised too, but Jacques Distler was easily able to solve the problem from scratch, and presumably Bolyai did the same.
Not that not knowing the transcendental stuff is an advantage, since it means that you’re not weighed down by the belief that the problem is insoluble. I saw a neat (true!) story about this on Snopes, where Dantzig (the linear programming guy) came late to class, saw two unsolved problems written on the board, took them for homework assignments and solved them.
They’re equivalent by definition. “Squaring the circle” means a finite algorithm (finite sequence of steps) for constructing a square of equal area to a given circle using only a compass and straight edge. That has been proven impossible, hence any algorithm must necessarily have an infinite number of steps.
The impossibility in Euclidean space arises because you have to construct a straight line of length pi from straight lines of rational length. Using a ruler and compass you can only create lengths that are sums, products, ratios, and square-roots of other lengths. But pi is “transcendental” (not in the buddhist sense), which among other things means it cannot be so constructed (proving the transcendentality of pi is difficult, and was not done for some time. Very few (non-artificially constructed) numbers have been proven transcendental, even though we know that “most” are).
I don’t think there is a big mystery why it works in the hyperbolic plane (H2). The formula for the area of a circle of radius r is 2pi sinh^2(r). The formula for the area of a quadrilateral is 2pi – (sum of interior angles).[that’s the weird thing about H2: polygon area depends on angles]. Since pi appears in both (unlike Euclidean where we have d^2 and 2pi r^2) , it seems reasonable that you can get one in terms of the other with rational operations and square roots.
WordPress interpreted certain symbols as formatting, plus I got the formula wrong.
That should be:
Area of circle of radius r in H2: 4 * pi * sinh(r/2) * sinh(r/2)
Ex-Anon —
I contest your use of “hence” in para 1 of your second post. In addition, equivalence is a 2-way relationship between statements, not 1-way.
I believe the following two statements are equivalent:
(A) Squaring the circle is impossible.
(B) EITHER (B1) there is no algorithm which squares the circle OR (B2) any algorithm which does square the circle has an infinite number of steps.
This is different from the statement in your first post, which I think is incorrect. Your second post only asserts that A implies B2, which is also incorrect. A implies (B1 OR B2); it is not the case that A implies B2.
And for equivalence, you would also need to show that B implies A.
While you might think this is pedantry, there is world of computational difference between no algorithm and one with an infinite number of steps.
JQ —
The story about Dantzig is relayed here. The problems had been posed by Jerzy Neyman, one of the founders of hypothesis-testing theory and a founder of modern mathematical statistics. I doubt Neyman thought they were impossible, just unsolved and hard.
Counter to your view, there is the experience of John Forbes Nash, who typically tackled unsolved problems from a state of ignorance, without reading any prior work, or leaning of other people’s attempts at solutions. Sometimes this worked in generating novel solutions (as with Nash equilibrium), but more often not.
Peter, I think x-anon is right on this one
Usage determines meaning, and in mathematical usage, handed down from the Greeks, “squaring the circle” means “describing a construction with straight-edge and compass, with finitely many steps, that yields a square with area exactly equal to that of a given circle”. This is impossible in the Euclidean plane.
As I’ve said before, I think the Nash approach accidentally adopted by Dantzig is a good one, at least to start with. Even if it doesn’t work, you will get a fresh view of the problem that may help you read the work of others.
I missed Jaque Distler’s link in jquiggin’s comment. There is a link to a Russian paper in his post, which solves the problem using the same approach I was suggesting (not that I was anywhere close, as reading the paper demonstrates, so I am glad I did not spend any more time on it 🙂
But you can’t do it for arbitrary circles. The sum of the interior angles of the square has to equal pi * w where w is rational [which makes complete sense since then the pi’s cancel between the circle and square areas].
My original comment that the number of steps must grow as the circle shrinks turns out to be correct, but the reasoning is more subtle than the “locally Euclidean” argument [Distler makes the same argument]. Just take w rational and arbitrarily close to 2. Then the square’s area is arbitrarily close to zero and hence so is the circle’s.
However, the number of steps required to construct the rational w = p/q (p,q coprime) is proportional to the size of p and q [roughly, you need q steps to split a line of unit length into q pieces of length 1/q, and then p steps to bolt p of the pieces back together to make a segment of length p/q]. As w gets closer to 2, p and q must grow without bound (eg, 3/2, 5/3, 7/4, 9/5….), and hence so does the number of steps in the algorithm.
JQ and ex-Anon —
We differ on the meaning which mathematicians place on the word “impossible” in this case. I believe common mathematical usage covers both the cases I identify as B1 and B2, not only B2.
On such tiny semantic arguments, entire disciplines are founded (eg, constructivist mathematics; Bayesian decision theory). 🙂
I am a bit doubtful about Distler’s method, because he does not provide a method for discovering/constructing the basic unit of his space. After all, if it was simply given down from on high, that is an extra tool to the straight edge and compasses. We can even square the circle (any circle) in Euclidean space if as well as compasses and straight edge we also have a suitable reference object, e.g. a right triangle with one short side 1 and the other pi.