4 ms·

How is it obvious? I am a mathematician and do not find it obvious in the least...

mikorym 7y ago
I also don't think it's obvious [1], but that it could seem obvious from the way some mathematics (ahem, calculus) is taught to students. [1] My first question will be whether your "circle" is of cardinality of R or of N... Edit: Actually, my question would rather be whether you allow countable or arbitrary union.
- rocqua 7y ago
  The statement included the word 'finitely many' thus excluding even countable union. At which point, the cardinality of the circle stops mattering I think. And at any rate, just... come on, we are obviously talking about a disk of radius r > 0 in the plane of real numbers using the euclidean metric.
BetaCygni 7y ago
As a non-mathematician, for a 2d circle on a flat plane: - The edge of a circle is curved (convex) - It does not matter how small a piece you cut of the edge, there will always remain a convex curve. - The edge of a square has straight edges, so you cannot put this piece at the edge. This means the old outside edge will have to be on the inside. - You cannot fit the convex edges to each other or to a straight edge. - Cutting a concave edge from the inside of the circle to fit the convex edge from the outside to will not help as it will produce a new convex edge. Ergo: there is no place to put the convex outside edge of the circle, so you cannot turn it into a square.
- deleted 7y ago
  [deleted]
- pfortuny 7y ago
  So not so OBVIOUS, anyway. You need the notion of convexity (not so easy to give abstractly unless you already know the definition). Items 4 and 5 look clear but from there to obviousness... Being visually intuitive is very different to being obvious.
  - authoritarian 7y ago
    This seems a bit pedantic. I'm certainly not a mathematician but it does seem obvious that you cannot cut a circle into any number of pieces and rearrange it to be a square, as a circle has curves which would prevent you from making the square a solid without gaps
    - pfortuny 7y ago
      No no; I’m trying to explain that “obviousness” is a bad idea to prove anything... Like “an infinite te has an infinite branch”...
- epanchin 7y ago
  You could create a concave edge with a series of triangles (really, trapezoids). Ps. I suppose this would really only work for an approximate circle, which is a polytope.
mrkeen 7y ago
I cannot visualise what to do with the (convex) curvy pieces that would be cut from the circle's border. There would be no concave pieces to slot them into - and I couldn't cut out new concavities for them without producing even more convex curves.
- pfortuny 7y ago
  Neither can I but... obviousness is very strong.
intuitionist 7y ago
It’s not obvious because, as stated, it’s wrong: https://arxiv.org/abs/1612.05833 https://arxiv.org/abs/1612.05833 You can prove that the disk and the square are not scissors-congruent, based on a convexity invariant, but if you allow more exotic dissections you can construct an equidecomposition.
- partyboat1586 7y ago
  Lots of equations no squared circle. As a picture book kind of person I'm disappointed.
rocqua 7y ago
Conservation of parts that have a curve.
aliceryhl 7y ago
No matter how you cut it, you will round edges that don't fit together with any other edge.

I also don't think it's obvious [1], but that it could seem obvious from the way some mathematics (ahem, calculus) is taught to students. [1] My first question will be whether your "circle" is of cardinality of R or of N... Edit: Actually, my question would rather be whether you allow countable or arbitrary union.

rocqua 7y ago
The statement included the word 'finitely many' thus excluding even countable union. At which point, the cardinality of the circle stops mattering I think. And at any rate, just... come on, we are obviously talking about a disk of radius r > 0 in the plane of real numbers using the euclidean metric.

BetaCygni 7y ago

As a non-mathematician, for a 2d circle on a flat plane: - The edge of a circle is curved (convex) - It does not matter how small a piece you cut of the edge, there will always remain a convex curve. - The edge of a square has straight edges, so you cannot put this piece at the edge. This means the old outside edge will have to be on the inside. - You cannot fit the convex edges to each other or to a straight edge. - Cutting a concave edge from the inside of the circle to fit the convex edge from the outside to will not help as it will produce a new convex edge. Ergo: there is no place to put the convex outside edge of the circle, so you cannot turn it into a square.

deleted 7y ago
[deleted]
pfortuny 7y ago
So not so OBVIOUS, anyway. You need the notion of convexity (not so easy to give abstractly unless you already know the definition). Items 4 and 5 look clear but from there to obviousness... Being visually intuitive is very different to being obvious.
- authoritarian 7y ago
  This seems a bit pedantic. I'm certainly not a mathematician but it does seem obvious that you cannot cut a circle into any number of pieces and rearrange it to be a square, as a circle has curves which would prevent you from making the square a solid without gaps
  - pfortuny 7y ago
    No no; I’m trying to explain that “obviousness” is a bad idea to prove anything... Like “an infinite te has an infinite branch”...
epanchin 7y ago
You could create a concave edge with a series of triangles (really, trapezoids). Ps. I suppose this would really only work for an approximate circle, which is a polytope.

mrkeen 7y ago

I cannot visualise what to do with the (convex) curvy pieces that would be cut from the circle's border. There would be no concave pieces to slot them into - and I couldn't cut out new concavities for them without producing even more convex curves.

pfortuny 7y ago
Neither can I but... obviousness is very strong.

intuitionist 7y ago

It’s not obvious because, as stated, it’s wrong: https://arxiv.org/abs/1612.05833 https://arxiv.org/abs/1612.05833 You can prove that the disk and the square are not scissors-congruent, based on a convexity invariant, but if you allow more exotic dissections you can construct an equidecomposition.

partyboat1586 7y ago
Lots of equations no squared circle. As a picture book kind of person I'm disappointed.

rocqua 7y ago

Conservation of parts that have a curve.

aliceryhl 7y ago

No matter how you cut it, you will round edges that don't fit together with any other edge.