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Complex Numbers on a Circle – Challenging Problem

My son is in 2nd grade. His math teacher gave the class a quiz, and one question was this:

If a triangle has 3 sides, and a rectangle has 4 sides,

how many sides does a circle have?

My first reaction was “0” or “undefined”. But my son wrote “$\infty$” which I think is a reasonable answer. However, it was marked wrong with the comment, “the answer is 1”.

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Is there an accepted correct answer in geometry?

**edit:** I ran into this teacher recently and mentioned this quiz problem. She said she thought my son had written “8” and didn’t know that a sideways “8” means infinity.

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The answer depends on the definition of the word “side.” I think this is a terrible question (edit: *to put on a quiz*) and is the kind of thing that will make children hate math. “Side” is a term that should really be reserved for polygons.

My third-grade son came home a few weeks ago with similar homework questions:

How many faces, edges and vertices do the following

have?

- cube
- cylinder
- cone
- sphere

Like most mathematicians, my first reaction was that for

the latter objects the question would need a precise

definition of face, edge and vertex, and isn’t really

sensible without such definitions.

But after talking about the problem with numerous people, conducting a kind of social/mathematical experiment, I observed something intriguing. What I observed was that

none of my non-mathematical friends and acquaintances had

any problem with using an intuitive geometric concept here,

and they all agreed completely that the answers should be

- cube: 6 faces, 12 edges, 8 vertices
- cylinder: 3 faces, 2 edges, 0 vertices
- cone: 2 faces, 1 edge, 1 vertex
- sphere: 1 face, 0 edges, 0 vertices

Indeed, these were also the answers desired by my

son’s teacher (who is a truly outstanding teacher). Meanwhile, all of my mathematical

colleagues hemmed and hawed about how we can’t really

answer, and what does “face” mean in this context anyway,

and so on; most of them wanted ultimately to say that a

sphere has infinitely many faces and infinitely many

vertices and so on. For the homework, my son wrote an explanation giving the answers above, but also explaining that there was a sense in which some of the answers were infinite, depending on what was meant.

At a party this past weekend full of

mathematicians and philosophers, it was a fun game to first

ask a mathematician the question, who invariably made various objections and refusals and and said it made no sense and so on, and then the

non-mathematical spouse would forthrightly give a completely clear

account. There were many friendly disputes about it that evening.

So it seems, evidently, that our extensive mathematical training has

interfered with our ability to grasp easily what children and

non-mathematicians find to be a clear and distinct

geometrical concept.

(My actual view, however, is that it is our training that has taught us that the concepts are not so clear and distinct, as witnessed by numerous borderline and counterexample cases in the historical struggle to find the right definitions for the $V-E+F$ and other theorems.)

I know I’m late to the party, but I’m surprised noone has mentioned this. In convexity theory, there is a notion called an extreme point that generalizes the notion of vertex (or corner) of a polygon. For this definition every point on a circle is an extreme point so it makes sense to say it has infinitely (uncountably!) many corners. Though the notion of side is not as good. If the definition is line segment joining two vertices then the answer would be 0 for the circle.

This is in reference to Douglas Stones’ answer, but images can’t be imbedded in comments. Limits of sides can have a straight angle, such as these octogons converging to a square.

A straight line could be any number of sides with straight angles between them.

For those who are thinking that the answer is $\lim \limits_{n \rightarrow \infty} n = \infty$, via:

- An $n$-gon has $n$ sides;
- A circle is a limit of a $n$-gon as $n \rightarrow \infty$;
- Therefore a circle has $\lim \limits_{n \rightarrow \infty} n = \infty$ sides;

I’d like to mention, it’s not so straightforward. If taking limits in this way were legitimate then we can show that e.g. a square has an infinite number of sides.

Consider a staircase with $n$ steps, and each step has height $1/n$ and width $1/n$. It consists of $2n$ line segments. As $n \rightarrow \infty$, the staircase converges to a single line segment (i.e. the limit agrees point-for-point with a single line segment).

If we glue four of these staircases together, and take their limit, we obtain a square, which would have $\lim \limits_{n \rightarrow \infty} 4 \times 2n = \infty$ sides.

Personally I use to think a circle had infinite sides as well, however, why could it not be one side with a $360^\circ$ curve?

I think the answer to this question relies heavily on the CW structure imposed on $S^1$. I can realise $S^1$ with an arbitrary number of $1$-cells.

Both answers 1 and $\infty$ are intuitively correct.

To the answer “$\infty$”:

Imagine that you start with circle. Now you can try approximate the circle by a centered (at middle of circle) hexagon. The next step is to double the number of corners to a regular dodecagon and so on. What you see geometrically is that the $n$-th regular polygon by this construction will approximate the circle better than the $(n-1)$-th one. You can look now at the number of sides during this approximation by doubling the number of corners:

$6\to12\to24\to48\to96…\to6\cdot2^n=3\cdot2^{n+1}$. Taking $n\to\infty$ you see that you get $\infty$ sides. (but their length goes to zero…)

To the answer “1”:

On the other hand it is not intuitive to call it a “side” while its length$\to0$, which is the state in a circle (remember the definition of a circle as a set of points). But what you get is a curved line (the circle itself), which one could interpret as a “side” because it separates the inner region from its environment. And this is **one** line. This could be the reason for the answer “circle has one side”.

However: “$\infty$ or 1?” is a question which causes from the question of the definition of the word “side”. (and as one can see “side” makes only really sense for polygons)

A circle has indeed $0$ straight sides.

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