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.
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.
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
It’s not obvious because, as stated, it’s wrong: 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.
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.
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.
Technically you could cut everything into infinitely many points and then reassemble them into another shape (especially given the volume is the same).
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.
If you accept the axiom of choice, then you can cut up a sphere into finitely many pieces and reassemble them to a sphere with twice the volume. (The pieces are not measurable, naturally.)
I think the line integral of curvature (changes sign based on concave / convexness) along perimeters remains the same under cutting / pasting; and it's different for squares and circles, so doesn't work out.
Seems like a nice approach, but is the line integral of curvature along the perimeter of a square defined? The corners are infinitely small line segments with infinite curvature.
Perhaps this works for squares with slightly rounded-off corners.
The square is defined as a union of lines, and the line integral of a whole shape is the summation of that of the curves it is a union of with open intervals not including the "corners" (points of 'infinite curvature' as you say). I think this definition would suffice because we just need to show that the sum is closed over cutting, no?
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.
https://dmsm.github.io/scissors-congruence/