Mathematicians Cut Apart Shapes to Find Pieces of Equations
quantamagazine.org
quantamagazine.org
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.
I bet there’s some other condition such as “small at infimum”, e.g. we can make as many cuts as we want we can’t really make infinitely many cuts.
(Alas, no, that will only give a countable set of cuts... but the image is pleasing unless you're a mouse)
However, if you have an approapriate metric (in this case I think cuts are area preserving) then you can't. [1]
[1] https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox#.... Look at the comment about measures changing.
Edit: Oh, I see you did say this at the end.
> "cannot cut a circle into finitely many pieces"
[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.
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.
- 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.
Being visually intuitive is very different to being obvious.
Like “an infinite te has an infinite branch”...
Ps. I suppose this would really only work for an approximate circle, which is a polytope.
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.
Perhaps this works for squares with slightly rounded-off corners.
You can eliminate a convex curvy bit by mating it with a concave curvy bit, but cutting out a concave curvy bit will again produce a convex curvy bit.
Can't get rid of the curvy bits!