How to distinguish walking on a sphere or on a torus?

http://math.stackexchange.com/questions/854380/how-to-distinguish-walking-on-a-sphere-or-on-a-torus

by ot • 12 years ago

7 3 12 years ago

math.stackexchange.com

If you're trying to distinguish between a "classic" torus (volume of revolution created by sweeping a circle around an axis coplanar with said circle) and a sphere, there's a fairly easy way.

Measure the curvature at your current point. (Say, by drawing a triangle and measuring angles). Walk in an arbitrary direction in a straight line until you come back to your starting point, measuring curvature at at least two points along the way. Then do the same at 90 degrees from your previous walk.

If you're on a torus, at least one of your measurements of the curvature will be different. If you're on a sphere, the curvature will be constant and positive.

socrates2014• 12 years ago

Walk two perpendicular loops. Try tightening the loops. On a torus, one will not tighten to a point. See Perelman and the problem he solved.

JadeNB• 12 years ago

How do you maintain your bearing? It's certainly possible to walk two contractible paths on a torus each of whose intersections are orthogonal. You can know that you've failed to keep your bearing if there are two distinct points of intersection between your paths, but I don't see how to guarantee that you succeed.

How to distinguish walking on a sphere or on a torus?

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