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Comment by mlyle | Hacker News Reader

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

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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.

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.

Then you must agree that the problem that was solved is not in the literal sense of the word "ancient."

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).

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.

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?"

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 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.

> 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?

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.

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.

> 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.

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.

Actually, a Banach-Tarski-like result is impossible in 2D space, since there is a Banach measure (= volume definition to all subsets of the plane) that extends the usual volume definition (e.g. for circles).

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.

maybe misleading but still interesting article . Maybe "new mathematics offers a solution to an Ancient problem"

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