# A set of points on a sphere

I found this interesting question, and I was wondering if anyone could help me out.

Let P be the set of points M on the earth with the property that if you go 7 miles North from M, then 7 miles West, and finally 7 miles South, you will find yourself back at the starting point M. Is P a closed set? If not, what is the closure of P?

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If you are 7 miles South from a polar circle whose circumference is 7 miles then you will arrive back at your starting point. You can extend this. Suppose that you are 7 miles South from a polar circle whose circumference is 7/n miles where $n$ is an integer, you can go 7 miles North to this circle, travel $n$ times around and go back 7 miles South. I'm not sure what the closure will be. – Aleks Vlasev Nov 11 '11 at 2:21
How do you go 7 miles North from the North pole? – Mariano Suárez-Alvarez Nov 11 '11 at 2:22
@AleksVlasev has correctly described the set of such points (with the exception of the South Pole itself). – Greg Martin Nov 11 '11 at 2:23

The "obvious" solution is the South Pole $S$. If you travel 7 miles north from $S$, it doesn't matter how much you travel west, you're still going to get back to $S$ once you go 7 miles south again.

The less obvious solutions are those points that are further than 7 miles from the north pole, and are such that after traveling 7 miles north, you lie on a latitude such that the circumference of that latitude is equal to $\frac{7}{n}$ miles, for a natural number $n$. If you travel 7 miles north (again, assuming that you can at all), and reach a point such that traveling 7 miles west has no ultimate effect, you will get back to where you started after going 7 miles south again.

As Mariano points out below, it would not even mean anything to "travel 7 miles north" if you are less than 7 miles from the north pole, so we must exclude these points explicitly to make sure the condition is specified for all points.

In other words, if $A_n$ is the latitude with circumference $\frac{7}{n}$, and $B_n$ is the set of points which, after traveling 7 miles north, you would reach $A_n$, then the points on $B_n$ are solutions.

$$P=\{S\}\cup\bigcup_{n\in\mathbb{N}}B_n$$

This is not a closed set; to produce the closure, you have to add the points that are exactly 7 miles below the North Pole - this corresponds to a circumference of 0, or "$B_\infty$". That is, $$\overline{P}=\{S\}\cup\left(\bigcup_{n\in\mathbb{N}}B_n\right)\cup B_\infty$$

Here is an extremely not-to-scale drawing:

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Also, I think we should note that $B_{\infty}$ itself is not a solution because there is no notion of "traveling 7 miles west" when you are at the North pole. – Aleks Vlasev Nov 11 '11 at 2:46
@Aleks: I think my answer makes it clear that $B_\infty$ isn't a solution, but I will try to make it clearer :) – Zev Chonoles Nov 11 '11 at 2:49
I would say that the definition of the set $P$ simply does not make sense, in that for points no further away from the North pole than 7 miles the condition for a point to belong to $P$ is meaningless —there is no sense in which one can say that they satisfy or not the condition, for the condition is meaningless for them. – Mariano Suárez-Alvarez Nov 11 '11 at 3:11
@Mariano: That's a good point, I've clarified my definition of $P$. – Zev Chonoles Nov 11 '11 at 3:24
Here's a follow up question that I have in mind that I'm not sure if I should post as a separate question. What would happen if "Earth" was a small ball, say, less than 7 miles in circumference. How would the solution change? – Aleks Vlasev Nov 11 '11 at 3:59