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If I have a circle and I start numbering points along the circumference with all the natural numbers: 1, 2, 3, 4, and so on, such that the length of the arc between two consecutive numbers is constant, what angle should be enclosed between the two radii of such consecutive points so that as I continue numbering points, I never have to number the same point twice?

Gentleman_Narwhal
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  • I might have not understood fully your construction, but if the arc length is constant, say $c$ then you will cover all the circle with a finite number of such arc lengths. Then you will inevitably starting counting the same points multiple times for each repetition. – MathematicianByMistake May 20 '16 at 12:20
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    The question is not about covering the circle with arcs, it is about numbering points on the circle. – Lee Mosher May 20 '16 at 12:26
  • @LeeMosher Thank you for the clarification. The way it is described, it can be interpreted like that too, but anyway.. – MathematicianByMistake May 20 '16 at 12:40

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Suppose that the angle separating consecutive numbers is $\alpha$. Without loss of generality, we can start numbering at $0$ and make it the reference angle $0$. Then $n$ would be at angle $n\alpha$, modulo $2\pi$. You never want to label the same point twice, which means that we require $$n\alpha \not\equiv m\alpha \pmod{2\pi}$$ for all $n\neq m$. Equivalently, the numbers $m$ and $n$ will be labelled at the same point if and only if there is some integer $k$ such that $$(n-m)\alpha = 2\pi k.$$ In other words, $\alpha$ is a rational multiple of $\pi$. It follows that any choice of $\alpha$ which is not a rational multiple of $\pi$ will never see the same point labelled twice.

EuYu
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  • But isn't $\pi$ irrational anyway? so would $\pi^2$ radians suffice? – Gentleman_Narwhal May 20 '16 at 12:39
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    $\pi$ is irrational yes, but that has nothing to do with $\alpha$. You need $\alpha$ to not be a rational multiple of $\pi$. So $\pi^2$ would be fine, as would $e\pi$ or $\sqrt{2}\pi$, but $2\pi$ or $3\pi$ or $5\pi/7$ would not, for example. – EuYu May 20 '16 at 12:43
  • Does the same apply if instead of marking natural numbers, I do the same but using all of the real numbers? – Gentleman_Narwhal May 20 '16 at 13:28
  • @Gentleman_Narwhal I'm not sure what you mean. In what sense are you using all of the real numbers? – EuYu May 20 '16 at 13:43
  • So for example the point of 1/2 would lie halfway (on the arc) between 1 and 2, the point $\pi$ would be found using the angle of $\pi\alpha$, etc. – Gentleman_Narwhal May 20 '16 at 14:10
  • @Gentleman_Narwhal I still don't fully understand what you're trying to do, but sing all the real numbers, you inevitably will have rational multiples of $\pi$. For example, no matter what angle $\alpha$ you pick, the point $\pi/\alpha$ and $3\pi/\alpha$ would lie at the same point. You seem some form of restriction to prevent stuff like this happening. – EuYu May 20 '16 at 22:29
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Hint: Think about when you will have to number the same point twice.

For example, say your angle is $x\cdot 2\pi$, where $x\in[0,1]$ (i.e., $x$ tells you what proportion of the 360 degre angle your angle is). That means you are numbering the points at angles $2\pi x, 2\pi\cdot 2x, 2\pi\cdot 3x,\dots$

Now, what is the condition that must be met in order for you to number the same point twice?

5xum
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  • Well, if the angle $(2\pi\cdot kx);mod;2\pi=0$ then I suppose you'd be back where you started...? – Gentleman_Narwhal May 20 '16 at 12:30
  • @Gentleman_Narwhal Well, not really. What you wrote down is the condition under which the starting point will be reused. Also, another hint... think about rational numbers and how a number $x$ is rational if and only if there exist two integers $p,q$ such that $q\cdot x = p$. – 5xum May 20 '16 at 12:33
  • I think EuYu's answer does a good job of explaining it further, and now I see where you're coming from, so does that mean that an irrational value of $x$ will mean that no points are ever reused? – Gentleman_Narwhal May 20 '16 at 12:45
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    You are correct! I didn't want to explain all of the details because my answer was the first, so I wanted to only nudge you in the correct direction. – 5xum May 20 '16 at 13:06