
Mathematicians Cut Apart Shapes to Find Pieces of Equations - pseudolus
https://www.quantamagazine.org/mathematicians-cut-apart-shapes-to-find-pieces-of-equations-20191031/
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mrcactu5
Here's a scissor's congruence app on github

[https://dmsm.github.io/scissors-congruence/](https://dmsm.github.io/scissors-
congruence/)

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rocqua
It seems obvious that you cannot cut a circle into finitely many pieces and
combine them into a square of the same area.

Hence, I suppose that in the starting statement of "If you have two flat paper
shapes and a pair of scissors, can you cut up one shape and rearrange it as
the other?", the concept "flat paper shapes" refers to polytopes [1]. That is,
N-dimensional generalizations of polygons.

Otherwise the statement "For two-dimensional shapes, the answer is easy: Just
determine their areas. If they’re the same, the shapes are scissors
congruent." is wrong with the circle - square case as a counter example.

[1]
[https://en.wikipedia.org/wiki/Polytope](https://en.wikipedia.org/wiki/Polytope)

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

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BetaCygni
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.

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pfortuny
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.

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authoritarian
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

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pfortuny
No no; I’m trying to explain that “obviousness” is a bad idea to prove
anything...

Like “an infinite te has an infinite branch”...

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doodpants
There's a fairly recent Numberphile video about this.

[https://www.youtube.com/watch?v=eYfpSAxGakI](https://www.youtube.com/watch?v=eYfpSAxGakI)

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mwexler
Didn't the topology folks figure some of this out a few years ago? Felt like
scissors congruent was just a restatement of some of the basics in topological
analysis, from what I (probably incorrectly) recall.

