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Could I define a circle as a polygon where the amount of sides approaches infinity? How would I define this mathematically?

Gerry Myerson
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Benichiwa
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  • Can you clarify what the purpose of this definition is going to be? – Brian Tung Apr 08 '15 at 00:58
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    It depends what you want to do with it. For measuring area, it's fine, but for things like measuring length, you have to be careful to avoid situations like considering a triangle as a stack of rectangles: this suggests the length of the hypotenuse is $a+b$, not $\sqrt{a^2+b^2}$. – Chappers Apr 08 '15 at 01:04
  • No, you can't define a circle as a polygon (regardless of whatever you may write after that). For some purposes, you may define a circle as the limit of a sequence of polygons. – Gerry Myerson Apr 08 '15 at 01:13
  • I wouldn't define it like that; it would be a terrible definition to work with! But you can show that the set of $n$th roots of unity, a prototypical regular $n$-gon, approaches the complex unit circle as $n \to \infty$. – pjs36 Apr 08 '15 at 01:16
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    @pjs36: In order for your statement to be true or false, you would have to define what "a regular $n$-gon approaches the complex unit circle as $n\to \infty$" means. – Jonas Meyer Apr 08 '15 at 02:17
  • @JonasMeyer That's very true; I would. Let's go with: "There's a radial bijection between the complex unit circle and points on the regular $n$-gon defined above. Call this family of bijections $\phi_n$. Then, for each point $z$ on the unit circle, the sequence ${\phi_n^{-1}(z): n\geq 3}$ converges to $z$ as $n \to \infty$." – pjs36 Apr 08 '15 at 02:34
  • @pjs36: So, pointwise convergence of a sequence of functions on $S^1$ to the identity function. This definition would leave out a lot of nice structure, e.g., the convergence of arclength and area which would hold for your example. – Jonas Meyer Apr 08 '15 at 02:37

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Yes, you could define the circle as the limit case of a sequence of regular polygons as you increase the number of edges. So in a certain sense, the circle is a regular “infinitogon”.

Mathematically you could e.g. compute the ratio between the circumcircle and the incircle of a regular $n$-gon. Then you take the limit $n\to\infty$ of that ratio and observe that it converges to $1$. So in the limit, circumcircle and incircle become the same. And since any regular $n$-gon is contained in the annulus between circumcircle and incircle, in the limit the $n$-gons have to converge to a circle.

MvG
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