Donut Problem!

Geometry Level 2

An ant crawls forward on a donut without turning.

If it crawls indefinitely like this, will it always return to its starting position at some point in time?

Note : Treat the donut as a perfectly smooth torus and the ant as a point. The ant can start at any point on the donut, and can choose any starting direction.

Yes No

Zach S. by Zach S. , 16, USA

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9 solutions

Zach S.
Jul 20, 2018

It won't always loop back. If you take a cylinder, and push the centers of the cylinders on the top and bottom together, then you get a Torus with the area being the lateral area of the cylinder. Now, if you take the cylinder, and take off the circles on the top and bottom, you can make the lateral area of the cylindar become a rectangle or a square. Now, on the Torus(donut), if you make a line, it will infinitely go on and loop around.

Same thing with the square now, if you make a line that goes on forever, then the line will just reappear on the opposite side of the square or rectangle being parallel to the original line from the same point just on the opposite side of the square. Now, say we start on one point one unit away from the perimeter of one of the sides. Then, the line starts moving at an angle and goes until it reaches the edge. It goes to the opposite end and continues. Now, it hits the edge again, and the distance between those two "edge points" is the square root of two. Since that number is irrational, it will never go back to one so the line won't always go back to its original spot.

P.S. my question got changed so the solution is for a point and a line.

36 Helpful 0 Interesting 0 Brilliant 0 Confused

will it go back to 1 if the circumference is the product of the square root of two?

Su Zx - 2 years, 10 months ago

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No because any rational and most irrational numbers(except other products of root 2) won't produce a rational number and the ant will still go around the donut without hitting any points that it passed through.

CHIN KEE HAW - 2 years, 10 months ago

Hm... it seems you are mixing up going straight on a flat torus (i.e. the topological torus with the flat metric) and going straight on the round torus (the topological torus with its round metric). In both cases the answer is the same, but it's not so trivial to show in the case of the round torus (it involves some non-trivial differential equations)

Antoine G - 2 years, 10 months ago

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I'm glad someone else noticed this. None of the explanations here work! The intuition from the flat torus is correct but it does not transfer directly. I don't think however, that differential equations are needed. Here is an intuitive outline, which hopefully can be made rigorous: Choose an arbitrary point on the torus and start walking. It is possible that the ant will at some point return to its original starting point with (importantly) the same velocity (i.e. direction of motion) as it had originally. If so, the path is said to be a ''loop''. Now note that loops form a discrete subset in the topology of paths from a given point. This is because if the initial angle is changed slightly, the path will no longer be a loop (this is the hardest part, not easy to justify). So there exist paths that are not loops. The points of self intersection of non-loops are at most countable in number. Therefore, choosing a point on the path that is visited only once will give a counterexample to the problem.

Chayim Lowen - 2 years, 10 months ago

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The idea sounds nice! but indeed, showing that there are some "non-loops" might be hard...

I think the differential equations are not that hard (at least in the "stable" set up). You only need a small perturbation of the "big loop" (the part of the torus which would touch the ground if you used it as a wheel). In the neighbourhood of that geodesic you might get a solvable differential equation. But (as there are no courses on surfaces or differential geometry on the website) setting up the differential equation seems to be outside the scope of the site.

Antoine G - 2 years, 10 months ago

There are periodic orbits and non periodic orbits. Therefore, the question is slightly confusing for us and not for ants as they will always be on periodic orbits.

Vinod Kumar - 2 years, 10 months ago

This "flow on a torus" is a classic example used in ergodic theory. The ant's path would be either periodic or ergodic, in which case its orbit is dense (ant gets arbitrarily close to every point on the torus, given enough time. The distinction is given by the rationality or irrationality of the ratio, which in the example is SQRT(2).

Will Heierman - 2 years, 10 months ago

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Be careful: the flow on the torus you speak is generally the linear flow. This means you are following a geodesic on a "flat torus". A donut, as one knows, is definitively not flat . This is the geodesic flow on the round torus which is way more complicated...

Antoine G - 2 years, 10 months ago

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True, but continuity and the intermediate value theorem indicate ergodic flows for "most" formulations of orbits that do not have periodic points. The example I cited is as you describe, and it is possible the ant would follow any of these paths. Therefore existence of ergodic orbits is assured, as is the negative answer to the question.

Will Heierman - 2 years, 10 months ago

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@Will Heierman – Correct. But do you know how to prove (with relatively simple arguments) that most geodesics on the torus are ergodic? I mean, this is definitively false for the sphere, so you need to show something.

Antoine G - 2 years, 10 months ago
Hu Yufan
Jul 30, 2018

The angle that the ant faces which deside how many small circle the ant could run once big circle.if it's a Irrational number,the ant would never back.

35 Helpful 0 Interesting 0 Brilliant 0 Confused

if the ration of the radii is irrational, then it might just so happen that an "irrational" angle is periodic. Also, you still have to prove that some geodesics are not loops. It could just so happen that all geodessics are loops.

Take the sphere for example, you take any angle you want, you will always come back to where you started.

Antoine G - 2 years, 10 months ago
Edwin Gray
Jul 30, 2018

The ant is a point. Therefore, on each loop the ant will cross one of an uncountable number of points. On the other hand, the cumulative number of crossing at each loop is countable. Since these are different orders of infinity, there can be no guarantee of returning to the same point. Ed Gray

14 Helpful 0 Interesting 0 Brilliant 0 Confused

You still have to prove that there are some geodesics which are not loops. If one apply your argument to the sphere, on sees there is something wrong. On the sphere all geodesics are loops, so the cross themselves an uncountable number of times.

Antoine G - 2 years, 10 months ago

Thanks. I had not understood what the question was because I was confused whether the line the ant was to travel was a straight line in the Euclidean sense, or a straight line in the donut space.

Kermit Rose - 2 years, 9 months ago
Arjen Vreugdenhil
Jul 31, 2018

The ant will, however, at least get arbitrarily close to its starting point.

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Alec Walter
Jul 29, 2018

someone can move the donut once the ant is off of it

3 Helpful 0 Interesting 0 Brilliant 1 Confused
Antoine G
Jul 31, 2018

Computing the geodesics on a "round" torus (the torus as represented in the image) is difficult. See here or here for the first steps into a proof.

See also here for nice animations (which show that the geodesics have nothing to do with straight line on a square)

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Ben Mitchell
Aug 2, 2018

The ant start on the outer circumference, Assume the outer edge of the donut has circumference 2 pi. Choose a point that is 1 away on the circumference. Have the ant return to that spot on its first loop. Then on the second loop it is at 2, etc. There are no two such integers N and M such that N x 1 will equal to any M x 2 x pi. else pi is rational.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

At first, the ant may look like that it'll return to its starting position but this isn't the case because of multiple curves in the donut. If the ant moves through the curves, the curves will cause the ant to turn, therefore, changing the direction that it is going so that it faces away from its starting point.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

This argument does not show in any way that the ant is not on a loop. There are plenty of geodesics on the round torus which are loops. You need to show that there is one geodesic which is not a loop.

Antoine G - 2 years, 10 months ago
Nia Bride
Jul 30, 2018

Imagine that an ant is painting a cut line for knife. You can cut donut in a such manner that you will not meet the starting point.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

The key words in the problem are "without turning". This is ambiguous. To me, that means the ant is confined to the intersection of the donut and a cutting plane. That intersection will define one or two ellipses. The ant must therefore return to its starting point.

Stephen Rasey - 2 years, 10 months ago

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