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