5 ms·
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
by rocqua 7y ago
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
- heavenlyblue 7y agoTechnically 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.
- jjgreen 7y agoIf 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.)
- pfortuny 7y agoAs they are not measurable, they do not possess a finite number of describing equations, so “cutting” is not a good analogy.
- klyrs 7y agoNah, just enchant a broom, cut it in two, and let them take care of the rest. (Alas, no, that will only give a countable set of cuts... but the image is pleasing unless you're a mouse)
- mikorym 7y agoAs sets of real numbers, yes. 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#Von_Neumann_paradox_in_the_Euclidean_plane 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.
- rocqua 7y agoFrom my post: > "cannot cut a circle into finitely many pieces"
- pfortuny 7y agoHow is it obvious? I am a mathematician and do not find it obvious in the least...
- mikorym 7y agoI 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 agoThe 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 agoAs 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 agoSo 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.
- nikki93 7y agoI 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.
- rocqua 7y agoSeems 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.
- nikki93 7y agoThe 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?
- failrate 7y agoMust the cuts be straight? If we allow curved cuts, it becomes possible despite your counterexample, doesn't it?
- rocqua 7y agoYou still have conservation of 'convex curvy bits'. 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!