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

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socrates2014
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.

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JadeNB
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.

