Have I made a straight line, or a circle?

Solution 1:

Though I (slightly) disagree with Nate's characterization of geometers in his comments, I think there is actually some deep notions worth discussing contained in it.

The heart of the matter is, how do you define a circle, and how do you define a line? And in the apparent paradox of the original question, the resolution is that there are (at least) two different ways of distinguishing circles versus lines, which, when taken together, allows for an object that is simultaneously a line in one definition but a circle in the other.

In the topological category, the circle and the line are the two examples of connected smooth one-dimensional manifolds. Since we are dealing with topology, everything is allowed to be stretchy/flabby, so the only difference we really care about between the circle and the line is that the latter is simply connected, while the former is not. More precisely, you can (trivially) find a closed loop in the circle which can not be continuously deformed to a point, while any closed loop on the line must "backtrack" sufficiently that it can be continuously shrunk down to a point.

A second description of the circle and the line comes from the geometrical category. (Here I'll just discuss the distinction between lines and non-lines.) A line in geometry is, intuitively, the straightest possible curve, which we take to mean "a curve that locally minimizes the distance between two points". It is an interesting special case of the Cartan-Hadamard theorem that, in a simply connected manifold of non-positve curvature (in particular the usual flat Euclidean space), rays (geodesics) emanating from the same point in two different directions will diverge forever and never intersect. (In fact, the divergence of geodesic rays is a characterization of non-positive curvature; also compare this to the case of positive curvature on a sphere, where any two great circles intersect at exactly two points.) So since the world in which we live our usual, everyday lives is more-or-less flat, our common intuition is that a line must be non-self-intersecting. Of course, if you consider traveling on the oblate spheroid that is the earth, you may come to a somewhat different conclusion.

The situation of Portal is precisely at the level of breaking the hypothesis of Cartan-Hadamard theorem in topology. As stated above, the Cartan-Hadamard theorem requires the space to be simply connected. By allowing the Portal, which is an identification two distinct subsets of usual Euclidean space, you pick up a non-trivial topology. And therefore Cartan-Hadamard theorem can fail. Hence your rope is allowed to be simultaneously a line in the sense that it is a straightest possible curve and a circle in the sense that it is a curve that returns to its starting point, and intrinsically has non-trivial topology.

Solution 2:

One way to capture the roundness of a circle is to embed it in a plane and measure distance between two points using the ambient plane. A different way to measure distance in a circle is to pretend you are an ant restricted to move within the circle. Then you measure the distance based on how far you walk in the circle. This is a "flat" circle. It is isometric to the circle depicted by the rope in your picture. (An ant walking on that rope couldn't tell any intrinsic difference between the rope and the circle in the plane.)