Suppose there's an ant which can do only two things: advance 1 unit forward and rotate 2π/n. For which integer values of n can this ant get arbitrarily close to any point in the plane?

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>!For every n except 1,2,3,4,6?!<
My feeling say that the larger n you have, the closer you can get to any point in the plane.
Every point in what universe? For irational numbers as cordinates the answear would be aproaching infinity.
I would start by showing that you can get arbitrarily point to any point iff you can make arbitrarily small points on the x- and y-axes. I would then try to find what scenarios let you construct these arbitrarily small points on the x- and y-axes. It should relate to particular ratios of trig functions being irrational, which we expect to happen most of the time. But certainly not all of it! You have an advantage setting this up for the x-axis because you get the vector (1, 0) for free; the y-axis is a little more difficult, for Reasons.
Can anyone make me understand this problem? What does "arbitrarily close to any point in the plane mean"?
n>0

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