There is an aspect of the word "construct" that comes off a bit sour here. There are infinitely many angles such that $\theta$ and $\theta/3$ are both constructable. However, the set of constructable angles $\theta$ for which $\theta/3$ is not constructable is dense. By continuity, it follows that there is no procedure for beginning with $\theta$ and using a finite number of compass and straightedge operations to actually find $\theta/3,$ even when it is constructable.

I wrote an article on the hyperbolic plane, constant curvature $-1.$ As pointed out by Bolyai, there are a countably infinite set of pairs, one circle and one square, such that the radius of the circle and the edge of the square are both constructable, and the two figures have the same area. However, there is no way to begin with one and find the other. The phrase "square the circle" is misleading, the reverse phrase "circling the square" would also be misleading if anyone ever used it.

I need to backtrack a little on that last phrase. If you have some sort of political plot going on in Moscow, Russia, and the hero is going around the perimeter of Red Square looking for the bad guys, it makes a good deal of sense to say that the hero is circling the square. The same would work with Times Square in New York.

Back to the Euclidean plane. Given an angle $\theta $ but not told what it is (in radians, say) there is no procedure for finding $\theta/3.$

There is an aspect of the word "construct" that comes off a bit sour here. There are infinitely many angles such that $\theta$ and $\theta/3$ are both constructable. However, the set of constructable angles $\theta$ for which $\theta/3$ is not constructable is dense. By continuity, it follows that there is no procedure for beginning with $\theta$ and using a finite number of compass and straightedge operations to actually find $\theta/3,$ even when it is constructable.

I wrote an article on the hyperbolic plane, constant curvature $-1.$ As pointed out by Bolyai, there are a countably infinite set of pairs, one circle and one square, such that the radius of the circle and the edge of the square are both constructable, and the two figures have the same area. However, there is no way to begin with one and find the other. The phrase "square the circle" is misleading, the reverse phrase "circling the square" would also be misleading if anyone ever used it.

I need to backtrack a little on that last phrase. If you have some sort of political plot going on in Moscow, Russia, and the hero is going around the perimeter of Red Square looking for the bad guys, it makes a good deal of sense to say that the hero is circling the square. The same would work with Times Square in New York.

Back to the Euclidean plane. Given an angle $\theta $ but not told what it is (in radians, say) there is no procedure for finding $\theta/3.$ Ever.

There is an aspect of the word "construct" that comes off a bit sour here. There are infinitely many angles such that $\theta$ and $\theta/3$ are both constructable. However, the set of constructable angles $\theta$ for which $\theta/3$ is not constructable is dense. By continuity, it follows that there is no procedure for beginning with $\theta$ and using a finite number of compass and straightedge operations to actually find $\theta/3,$ even when it is constructable.

I wrote an article on the hyperbolic plane, constant curvature $-1.$ As pointed out by Bolyai, there are a countably infinite set of pairs, one circle and one square, such that the radius of the circle and the edge of the square are both constructable, and the two figures have the same area. However, there is no way to begin with one and find the other. The phrase "square the circle" is misleading, the reverse phrase "circling the square" would also be misleading if anyone ever used it.

Back to the Euclidean plane. Given an angle $\theta $ but not told what it is (in radians, say) there is no procedure for finding $\theta/3.$

There is an aspect of the word "construct" that comes off a bit sour here. There are infinitely many angles such that $\theta$ and $\theta/3$ are both constructable. However, the set of constructable angles $\theta$ for which $\theta/3$ is not constructable is dense. By continuity, it follows that there is no procedure for beginning with $\theta$ and using a finite number of compass and straightedge operations to actually find $\theta/3,$ even when it is constructable.

I wrote an article on the hyperbolic plane, constant curvature $-1.$ As pointed out by Bolyai, there are a countably infinite set of pairs, one circle and one square, such that the radius of the circle and the edge of the square are both constructable, and the two figures have the same area. However, there is no way to begin with one and find the other. The phrase "square the circle" is misleading, the reverse phrase "circling the square" would also be misleading if anyone ever used it.

I need to backtrack a little on that last phrase. If you have some sort of political plot going on in Moscow, Russia, and the hero is going around the perimeter of Red Square looking for the bad guys, it makes a good deal of sense to say that the hero is circling the square. The same would work with Times Square in New York.

Back to the Euclidean plane. Given an angle $\theta $ but not told what it is (in radians, say) there is no procedure for finding $\theta/3.$