How to prove the infinite number of sides in a circle?

Solution 1:

In a way, there is no reverse way for you to prove. You must always be careful about what your definitions. That way, you will always have a clear mathematical way of writing what you are trying to prove and how to prove it, as well as how "reverse ways" of some theorems would look.

For example, in your case, you have defined the following:

• regular polygons with $n$ sides.
• The angles of these polygons.

What you have shown is this:

As $n$ approaches infinity, the external angle of a $n$-sided regular polygon approaches $0$.

What you Have not shown is this:

As $n$ approaches infinity, the $n$-sided regular polygon approaches a circle.

Why haven't you shown this? Well, let's see:

First of all, you haven't defined a circle. Sure, you can define a circle, but a circle will not be a regular polygon, at least not by the definition of a regular polygon. OK, that may be a problem you can overcome. We can say that a circle is some sort of curve, just like a polygon. However, there is another problem:

You did not define what it means for two curves to approach one another. Without defining exactly what it means for a polygon to approach a circle, you cannot say a circle approaches it, neither can you then say "How to prove the inverse of this statement?"

Even if you define what it means for two curves to approach each other, you are still miles away from showing the statement which you, ultimately, want to show:

A circle is a regular polygon with an infinite number of sides

Again, why could you not proven this statement? Well, it isn't a mathematical statement, so it cannot be proven in a mathematical way. For example, a polygon is defined as a collection of a finite amount of straight lines, so the concept of "a regular polygon with an infinite number of sides" does not exist yet. You can define it, sure, but if you just define it as a circle, then the statement becomes empty. You could define "generalized polygons" as such:

A curve is a generalized polygon if it is a limit of a sequence of (finite-sided) polygons.

In this case, you must, of course, define what a limit of a curve is, but that is possible (albeit not trivial).

If you decide to define it that way, you can now prove the statement:

If $P_n$ is a regular $n$-sided polygon, then the limit of the sequence $P_1, P_2, \dots$ is a circle.

This statement is, basically, the statement "$n$-sided polygons approach a circle as $n$ tends to infinity", in mathematical terms. However, notice what happened:

• You can no longer speak of a "reverse" of this statement. The statement, by its nature, works only in one direction: the limit of this sequence is this curve.

• In defining generalized polygons, you lost the ability to speak about the number of sides of a polygon. For example, a square is a $4$ sided polygon and has $4$ sides. You can say "the number of sides of this polygon is such and such". You cannot say the same thing about generalized polygons. You can define the number of sides as such:

  The number of sides of a generalized polygon $P$ is $n$, if $P$ is a $n$-sided polygon, and is $\infty$ if, for all values of $n$, $P$ is not a $n$ sided polygon.

If you decide the number of sides that way, then the statement

A circle has an infinite number of sides

Becomes equivalent to the statement

A circle is a generalized polygon and for all values of $n$, a circle is not a $n$ sided polygon.

Solution 2:

For all intents and purposes, infinity is not a number. It doesn't really make sense to say $n = \infty$. The integers have order, and if it were true that $n = \infty$ then we would get the following logic: $$\infty = n < n+1 = \infty \implies n = n+1$$ This is why a limit is useful (as you have denoted with the notation $n \to \infty$) because we can explore what happens to an expression if we allow $n$ to be as large of an integer as we want. A large enough number will cause our expression to behave similarly to infinity, which is why we know things like $\lim_{n \to \infty} \frac{360^\circ}{n}=0$. This limit is true in an algebraic sense, but geometrically we don't really make sense of an infinite sided polygon with an exterior angle of zero. After all, that would imply that $\infty \cdot 0 = 360^\circ$, as the sum of all the exterior angles must add to $360$ degrees. But is it possible that $\infty \cdot 0 = 360$? How do we know if instead it should be $\infty \cdot 0 = 0$ or $\infty \cdot 0 = \infty$? These are questions you will need to explore with calculus, where you can develop a much stronger notion of the limit. With the aid of calculus, you may be able to reverse-engineer the problem you are trying to work on.

Anyway, here is what I think is the main problem you are running into. The formula you have to calculate the external angle of a regular polygon is true for a polygon with $n$ sides. So, if you define a circle as "the limit of a regular polygon with $n$ sides", you cannot apply that formula because it is only true for those regular polygons with finitely many sides. As we've established, $n \neq \infty$, so you simply cannot use this formula for a polygon where $n = \infty$

Solution 3:

Here is my approach to a proof that a circle has infinitely many sides:

• We consider an equilateral n-gon. The n vertices are equidistant from the center.
• If n is infinity, there is an infinite number of vertices that are equidistant from the center. That is the definition of a circle.
• For each adjacent pair of vertices on the n-gon, there is a side (edge) connecting them. Therefore, if there is an infinite number of vertices, there would have to be an infinite number of sides.