5 ms·

I almost feel bad making such a trivial point in response to such a nontrivial article, but they're not solving the ancient problem of doing it with a compass a

I almost feel bad making such a trivial point in response to such a nontrivial article, but they're not solving the ancient problem of doing it with a compass and a straightedge (which was proven impossible in the late 1800s), they're solving another problem that shares the similarity that there's a square and a circle.

mlyle 5y ago
That's well explained in the article: > 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
- gus_massa 5y ago
  The problem is that the title is misleading. It's like an article with the title "New world record in the 100 meters dash using a new technique" and in the middle of the article it explains that it's about bungee jumping a 100m fall.
  - mlyle 5y ago
    The squaring a circle with compass and straightedge has been known to be impossible for 140 years. We're obviously talking about the version of the problem everyone's been working on for roughly the last 100.
    - whatshisface 5y ago
      Then you must agree that the problem that was solved is not in the literal sense of the word "ancient."
      - mlyle 5y ago
        This is just a further result: - 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).
    - ruggeri 5y ago
      It might be obvious to people who are familiar with the problem. In fact, the title confused (and baited) me because I was familiar with the problem, but not the new formulation.
  - dfabulich 5y ago
    I think it's fair to say that this is an "ancient problem." The ancients only had a compass and straightedge, but they were asking a general question, "can you square the circle?" They weren't asking "can you square the circle using only these two tools currently known to us?"
    - gus_massa 5y ago
      The old problem is solvable with a compass, a straightedge and a rope. [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 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.
- paulpauper 5y ago
  maybe misleading but still interesting article . Maybe "new mathematics offers a solution to an Ancient problem"
- thaumasiotes 5y ago
  > 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 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?
  - OscarCunningham 5y ago
    Banach-Tarski doesn't work in 2-dimensional space; there isn't a finite collection of subsets of the plane which can be assembled to make both one disc of radius one and two discs of radius one. I believe that Banach-Tarski would make it much easier to disect a sphere and make a cube.
  - ummonk 5y ago
    Chopping and rearranging something that's the same spherical shape (but different size) is different from chopping a 2D square and rearranging into a circle. Presumably, if it were easy, Tarski himself would have shown it, given that he's the one who posed the question.
    - thaumasiotes 5y ago
      > Chopping and rearranging something that's the same spherical shape (but different size) is different from chopping a 2D square and rearranging into a circle. 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.
  - mlyle 5y ago
    As other people point out, 2-space is not 3-space. Only weakened version of B-T work in 2-space (e.g. infinite number of pieces). Tarski, of course, was familiar with his work of the year before formulating the B-T paradox when he posed this question.
jnsie 5y ago
I agree with you completely. That it was explained in the article, does not change the article's title/premise.
HotHotLava 5y ago
I think this is overly narrow: Anaxagoras considered the problem of turning a circle into a square of the same area, and these mathematicians shed new light on that problem. He was probably thinking of compass and straightedge because it was the only language he had for attacking the problem, but it's not he published a paper with the precise definition of the terms and the theorem he tried to prove. 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.
- thaumasiotes 5y ago
  https://en.wikipedia.org/wiki/Squaring_the_circle#History https://en.wikipedia.org/wiki/Squaring_the_circle#History > It is believed that Oenopides was the first Greek who required a plane solution (that is, using only a compass and straightedge).
deleted 5y ago
[deleted]
analog31 5y ago
I took an abstract algebra course in college. I don't think we talked about squaring the circle, but we did visit the problem of trisecting the angle. What we learned is that compass-straightedge construction can execute a finite number of primitive operations, notably the arithmetic operators +, -, *, and /, along with square root. And that's it. Then we showed that trisecting the angle requires a cube root button, which is what makes it impossible. The cool thing was finding a correspondence between geometry and algebra, followed by completing the proof as an almost trivial algebra problem.

mlyle 5y ago

That's well explained in the article: > 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

gus_massa 5y ago
The problem is that the title is misleading. It's like an article with the title "New world record in the 100 meters dash using a new technique" and in the middle of the article it explains that it's about bungee jumping a 100m fall.
- mlyle 5y ago
  The squaring a circle with compass and straightedge has been known to be impossible for 140 years. We're obviously talking about the version of the problem everyone's been working on for roughly the last 100.
  - whatshisface 5y ago
    Then you must agree that the problem that was solved is not in the literal sense of the word "ancient."
    - mlyle 5y ago
      This is just a further result: - 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).
  - ruggeri 5y ago
    It might be obvious to people who are familiar with the problem. In fact, the title confused (and baited) me because I was familiar with the problem, but not the new formulation.
- dfabulich 5y ago
  I think it's fair to say that this is an "ancient problem." The ancients only had a compass and straightedge, but they were asking a general question, "can you square the circle?" They weren't asking "can you square the circle using only these two tools currently known to us?"
  - gus_massa 5y ago
    The old problem is solvable with a compass, a straightedge and a rope. [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 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.
paulpauper 5y ago
maybe misleading but still interesting article . Maybe "new mathematics offers a solution to an Ancient problem"
thaumasiotes 5y ago
> 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 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?
- OscarCunningham 5y ago
  Banach-Tarski doesn't work in 2-dimensional space; there isn't a finite collection of subsets of the plane which can be assembled to make both one disc of radius one and two discs of radius one. I believe that Banach-Tarski would make it much easier to disect a sphere and make a cube.
- ummonk 5y ago
  Chopping and rearranging something that's the same spherical shape (but different size) is different from chopping a 2D square and rearranging into a circle. Presumably, if it were easy, Tarski himself would have shown it, given that he's the one who posed the question.
  - thaumasiotes 5y ago
    > Chopping and rearranging something that's the same spherical shape (but different size) is different from chopping a 2D square and rearranging into a circle. 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.
- mlyle 5y ago
  As other people point out, 2-space is not 3-space. Only weakened version of B-T work in 2-space (e.g. infinite number of pieces). Tarski, of course, was familiar with his work of the year before formulating the B-T paradox when he posed this question.

jnsie 5y ago

I agree with you completely. That it was explained in the article, does not change the article's title/premise.

HotHotLava 5y ago

I think this is overly narrow: Anaxagoras considered the problem of turning a circle into a square of the same area, and these mathematicians shed new light on that problem. He was probably thinking of compass and straightedge because it was the only language he had for attacking the problem, but it's not he published a paper with the precise definition of the terms and the theorem he tried to prove. 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.

thaumasiotes 5y ago
https://en.wikipedia.org/wiki/Squaring_the_circle#History https://en.wikipedia.org/wiki/Squaring_the_circle#History > It is believed that Oenopides was the first Greek who required a plane solution (that is, using only a compass and straightedge).

deleted 5y ago

[deleted]

analog31 5y ago

I took an abstract algebra course in college. I don't think we talked about squaring the circle, but we did visit the problem of trisecting the angle. What we learned is that compass-straightedge construction can execute a finite number of primitive operations, notably the arithmetic operators +, -, *, and /, along with square root. And that's it. Then we showed that trisecting the angle requires a cube root button, which is what makes it impossible. The cool thing was finding a correspondence between geometry and algebra, followed by completing the proof as an almost trivial algebra problem.