> Because a previous result had demonstrated that it’s impossible to use a compass and a straightedge to construct a length equal to a transcendental number, it’s also impossible to square a circle that way.
> That might have been the end of the story, but in 1925 Alfred Tarski revived the problem by tweaking the rules. He asked whether one could accomplish the task by chopping a circle into a finite number of pieces that could be moved within a plane and reassembled into a square of equal area — an approach known as equidecomposition
We're obviously talking about the version of the problem everyone's been working on for roughly the last 100.
- You can't do it with a compass and a straightedge, like the ancients were trying (1880's?)
- You can't do it with scissors, either (mid-20th century?)
- But with modern mathematics, and really complicated shapes-- you can. (now).
They weren't asking "can you square the circle using only these two tools currently known to us?"
[You wrap the rope around the circle and straight it to get a segment of length 2πR, and take the middle point to get a segment of length πR. Then use the compass to continue it with a segment of length R. And then calculate the square root like in https://www.geogebra.org/m/edtecfcv to get a segment of length sqrt(π)R that is the side of your square. I'm sure this was known in ancient times.]
Using only compass and straightedge is more like a esthetics decision.
The old problem is difficult (impossible) because you have strong restrictions about which points you can draw. You have no rope and no magic rule to get any arbitrary length.
The new problem is difficult because you must cut one figure and rearrange the parts to get the other figure.
They have very different restrictions, in spite both are about a circle and a square with the same area.
Huh. The Banach-Tarski theorem ("you can chop a sphere into a finite number of pieces and, by moving them within 3-space, reassemble them into a sphere of double the radius") strongly suggests this is possible. What's so interesting about the revised question?
I believe that Banach-Tarski would make it much easier to disect a sphere and make a cube.
As the other responses point out, the Banach-Tarski construction uses properties of 3D space that do not occur in 2D space. But I don't think the "spherical shape" of the original volume or the final volume is relevant; the point of the construction is that the intermediate pieces can't really be said to have a shape.
Tarski, of course, was familiar with his work of the year before formulating the B-T paradox when he posed this question.
The crucial idea that makes Banach-Tarski work in 3D is the insight that the set of rotations around an axis through the origin in 3-space has a free subgroup F on 2 generators (finite strings of A's, B's and their inverses). From this fact the proof is quite easy, but this comment is too small for it.
From what I can tell from a cursory search, there is no surviving fragment concerning squares and circles from Anaxagoras himself, and the mention on Wikipedia goes back to a quote from Plutarch:
> There is no place that can take away the happiness of a man, nor yet his virtue or wisdom. Anaxagoras, indeed, wrote on the squaring of the circle while in prison.
> It is believed that Oenopides was the first Greek who required a plane solution (that is, using only a compass and straightedge).
The cool thing was finding a correspondence between geometry and algebra, followed by completing the proof as an almost trivial algebra problem.