Question

a curve is totally contained inside the square with vertices (0,0), (1,0), (1,1), and (0,1). is there any limit to the possible length of the curve? explain.

Answer 1

are there any limits on the shape of the curve?

Answer 2

@TuringTest not that im aware of.

Answer 3

we can argue this a few ways then, but I suggest trying to approach this from a point of contradiction. If there is a longest curve, what would it be like?

Answer 4

a u-shape i'd suppose, that would be the only possible curve i could think of that would fully use the space given inside of the square. @TuringTest

Answer 5

many such u-shapes, right?
and in order for it to "fully use the space", how close would those curves have to be to each other?

Answer 6

infinitely close.

Answer 7

exactly, so how many such u-shapes fit in the picture then?

Answer 8

there is no way to tell, so an infinite amount? therefore, there is no limit to the possible length?

Answer 9

right, if we need the shapes to be infinitely close, then we need infinitely many turns, which is infinitely many u-segments. since each segment has a length, and the lengths are not tapering off to some limit, they must keep adding, hence they make an infinitely long curve.

Answer 10

the question is equivalent to asking something like "if you unravel a square with side 1 completely, how long is it?"
this is even more obvious, since you are unraveling an area to get a length, and they have different untis.
also equivalent:
how many points are on the line segment [0,1] ? again we can cram infinitely many such points.

Answer 11

thank you so much for explaining!

Answer 12

welcome, it's an interesting question :)