# Have I made a straight line, or a circle?

(Disclaimer: I'm an engineer)
Hi everybody, I found this “riddle” posted on the internet:

It's meant as a joke, but I do think it deserves an answer :)
A bit of background: the orange and blue ellipses are a citation from the videogame Portal. They are "portals" connected so that everything that goes into one of them comes out of the other, mantaining his momentum (this last part is the foundation for the game). For further info, you can also watch the trailer.

So.. is it a circle, a straight line or what else?

(also, feel free to retag)

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A circle with large enough radius looks like a straight line up close. Why did ancient people think the world was flat? – J. M. Dec 7 '10 at 14:22
If you're a topologist, you've made a circle. If you're a differential geometer, you've made a line. – Nate Eldredge Dec 7 '10 at 14:28
It can't be a circle with that knot in the middle. Circles don't have any distinguished points. – Ross Millikan Dec 7 '10 at 14:32
@Nate: I disagree. In differential geometry one can accomodate both notions at the same time. I'd say he made a closed geodesic, modulo Ross's objection. – Willie Wong Dec 7 '10 at 14:35
@J.M.: Except that you typically expect circles of infinite radius (zero curvature) to have infinite length, which is not the case here. – Nate Eldredge Dec 7 '10 at 15:20

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.

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To summarize this excellent answer: you have a geodesic which is also a circle. Geodesics generalize the notion of a line, and it's because of situations like this where geodesics aren't lines that we need to introduce new precise terminology like "geodesic." – Noah Snyder Dec 7 '10 at 18:16
Very thorough and clear enough that I could get most of it :) Also, since nobody came here bashing you in the comments, I assume it is correct ;) – Agos Dec 8 '10 at 21:49

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.)

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Is the equator of the Earth a line or a circle?

Well, it's a circle of course. But as you know by now, from Willie Wong's answer and from others, it's also a geodesic, which is very much like a straight line: unless you are ready for some serious digging, the equator is the shortest path between any two places on it.

Your Portal picture is nothing like the Earth, mind you. But in my kitchen I have a cutting board that's a flat rectangle with a thin rectangular hole cut out near one side, as a handle for carrying. The corners are rounded somewhat. Now an ant walking along the top side, if it passed through the handle, would wind up on the bottom side with its momentum reflected. The same if it walked over an edge.

Imagine an orthographic projection of my cutting board. (By the way I'm also an engineer. I learned drafing the old-school way, with a T-square.) I draw a top view which is a rectangle with a thin vertical rectangular hole near the right side. To the right I place a side view which is just a vertical rectangle with a couple of hidden lines, the thickness of the cutting board. It's not normal "third-angle" to also draw the bottom view, but I will, so it is a rectange with a thin vertical rectangular hole near the left side.

An ant walking over the right edge would, on the drawing, pass from the top view to the side view to the bottom view with its momentum unchanged. (From a 2-D perspective, the edges are not actually special. The corners are another matter.) If it walked through the handle, it would pass from one rectangle on the top view through an "interspace" - hidden lines on the side view - to the other rectangle on the bottom view, again with its momentum again unchanged. Imagine if we tied a string through around the handle and the right edge of the cutting board. On the drawing, it would be a straight line between the two rectangles passing through the three views.

So the area around the handle and right edge of my cutting board is very much like a 2-D version of Portal. 3-D Portal can then be thought of as being played on the hypersurface of a higher dimensional cutting board. (To visualise a higher dimensional cutting board, ponder an ordinary cutting board while enjoying a "herbal cigarette".)

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Visualizing multidimensional geometry while on "herbal cigarettes" sounds like a fun time. :) – J. M. Dec 11 '10 at 15:18
Try to picture what the "corners" would look like to someone in that space. Duuude. Hmm, I'm hungry. – amateur topologist Dec 11 '10 at 19:26

## protected by Asaf KaragilaDec 14 '13 at 22:23

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