It’s a primary school homework question which causes disagreement among seasoned mathematicians. In order to find the correct answer, I called upon that most rigorous of scientific instruments, the Twitter poll:

In retrospect I regret not including a “none of the above” option, but more on that later. In this post I’m going to go through these three answers (and “None of the above”), and discuss their pros and cons as I see them, before dramatically revealing the correct response.

But first: why can’t we straightforwardly give the right answer? The words in the question are hardly mysterious. We all know what a “circle” is, what it means to count “how many” of something, and what a “side” is… or don’t we? Here are (rough) definitions I distilled from discussions with two primary school students who had been on the receiving end of this question:

1. A line forming part of a shape’s boundary of a plane figure.

The purpose of saying plane figure rather than “shape” here is that we want shapes that live in 2-dimensions (e.g. squares or circles, but not spheres or cubes). The next question is what a “line” is in definition 1. Here’s a variant which pins that down:

2. A straight line forming part of the boundary of a plane figure.

If you type “define: side” into Google, the most relevant definition is:

A rectilinear figure is one constructed from straight lines. So this definition is a further refinement of definition 2, and allows us to affirm that a square has four sides, but on the face of it has nothing to say about non-rectilinear plane figures such as circles.

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Infinitely Many Sides?

I think it’s a safe bet that the respondants to my Twitter poll have a higher level of mathematical education than the national average. The fact that they were split on this question at all, and that a small majority selected an answer which is pretty well unavailable to this question’s usual audience (primary school students), certainly suggests something going wrong somewhere.

So, does a circle have infinitely many sides? It is definitely useful to consider a circle as the limit of n–sided polygons as n gets bigger and bigger. This is exactly the approach that Archimedes, Liu Hui, and countless others have used down the centuries to study circular geometry, including coming up with approximations for π.

Sometimes it is absolutely sensible, as a convenient shorthand, to think of a circle as being like a polygon with infinitely many sides.

But, as an insufferable pedant a mathematician, I’d want to distinguish between convenient shorthand and literal truth. If we’re adamant that a circle really is a polygon with infinitely many sides, then the question presents itself: what are the sides? And surely the only plausible answer^[1] is: the individual points of the circle. How long are these so-called sides? Zero centimetres. And are these sides separated by corners? Not apparently, either there are no corners at all, or every point is both a side and a corner.

I’d say sides of length zero are… a problematic concept.  How can you tell whether you have got any? For instance, suppose I’m studying a system where a square arises as a limit of octagons like this:

In this situation, it might well make sense for me to think of my square as having eight sides, four of which have length zero. But if I was to insist that my (perfectly ordinary) square really does have eight sides, you might raise an eyebrow.

So this – the winning answer in my poll – is the only one I am going to declare to be definitively wrong, while it is also the only one which offers any geometrical insight at all. A paradox? Not really. Reasoning by analogy is a valuable skill in mathematics and in life; at the same time it is important to hold on to the realisation that that is what we are doing.

For infinitely many sides: geometrically illuminating.
Against infinitely many sides: eight-sided squares.

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Off on a tangent 1: apeirogons

Even if a circle isn’t one, are there such things as polygons with infinitely many sides? Well, there is a word to describe such a thing an apeirogon. A regular apeirogon would then have sides of equal (non-zero) length with equal angles between them. The only option here is this stupendously unexciting object:

If you object to this as being a “polygon” (either bcause of the angles of 180° or the chain of edges not closing in a loop), how about something like this: start at the bottom of a circle, and at each stage move around half of what remains of the circle, and replace the arc you’ve just travelled with a straight edge:

Is this a genuine polygon? Once again it depends on your terms. According to one common definition, that of a “closed polygonal chain”, this fails to qualify since the starting corner (bottom left) only connects to one edge. But it’s a very near miss: that point is the limit of a sequence of edges from the right, making this shape a “non-self-intersecting piecewise linear closed curve”, another definition of polygon that people use.

If we leave our usual Euclidean world and enter hyperbolic space, then there is no ambiguity. Apeirogons (even regular apeirogons) simply exist:

Off on a tangent 2: extreme points

It might be more defensible to say that a circle has infinitely many corners than infinitely many sides (although this is not a question that seems to get asked very often). To start with, if a corner of a square is a point at which its boundary line is not straight, then every point on the circle satisfies that. More sophisticatedly, there is a notion of an extreme point of a shape: that is any point through which you can draw a segment of straight line which touches the shape only at that exact point. For a square and many familiar shapes the extreme points exactly coincide with the corners. Every point on the boundary of the circle is an extreme point, so it is certainly true that a circle has infinitely many.

We might worry that some shapes such as this chevron have corners which are not extreme points:

Here the lower central corner is not an extreme point (the other three corners are). What’s going wrong is that this shape is not convex (roughly, it has some bits sticking out too far). A circle is convex, so perhaps we needn’t worry. Alternatively, we might remedy the situation by defining a “corner” to be a point which is an extreme point either of the shape in question or of its complement, i.e. the whole plane with the shape cut out of it. That approach would detect corners of all polygons, including the chevron. For smooth curves, it would identify all boundary points as “corners” except for inflection points (which is not unreasonable since we might argue that the boundary is straight there).

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One Side?

In primary school, it seems that “one” is the answer that gets the tick. And there is a moderately decent justification. Remember definition 1 above:

1. A line forming part of the boundary of a plane figure.

The immediate question is what counts as a “line”, especially if we are not insisting on straightness. If we are too relaxed about this, then any plane figure could be said to have “one side”, in the same sense that it has one boundary, perimeter, or circumference. But this has got to be wrong, as we surely want a square to have four. Well, a square has four points where it is not smooth, with four smooth sections in between. Perhaps it was really the smooth sections that we were counting all along. So implicitly we have a new refinement of definition 1 (and also take the opportunity to ditch the vague term “figure”):

4. Each smooth section of a piecewise-smooth closed curve.

A “closed curve” is one which loops around to meet itself so that it has no free ends. “Piecewise-smooth” means that it is built from smooth sections, which meet at isolated unsmooth points. It is perfectly legitimate to want to count the smooth sections of such a shape’s boundary, and it’s by no means outrageous to use the word “side” when doing that. So I’m certainly not saying this is definitively the wrong answer.

The question is whether that interpretation of “side” is not merely coherent, but natural enough that it can simply be presumed without being stated explicitly (which it seldom if ever is). What happens when smoothness and straightness tally up differently? Consider this tombstone shape, created by replacing the top of a square with a semicircle of equal diameter.

This has two smooth sections (the bottom line and the rest) but three straight edges (plus a curved piece smoothly^[2] joining two of them). So how many sides does it have? I consulted my Twitter friends again:

This time I should have included “infinitely many” as an option, although that can be absorbed into “None of the above”. Anyone voting that the circle has infinitely many sides should automatically vote “None of the above” here, unless – an important caveat – the nature of this shape indicates to the reader a different notion of “side”. The fact that the most popular choices in these two polls are incompatible suggests this may be the case (or at least reinforces that the waters are muddy).

Although two is a perfectly respectable answer, compatible with definition 4 above and with a circle’s one-sidedness, I am not satisfied it is definitively the right one, or that three or four are categorically wrong. It depends what you want to count: smooth sections, straight edges, or straight edges plus whatever’s then leftover, any of which might be the answer you want depending on context (more on this below). Relatedly, I am not sure counting the number of smooth sections fully matches my intuition of the word “side”. After all, the tombstone’s two upright sections are – I think it is fair to say – “on opposite sides”. Are we really content that they are simultaneously part of the “same side”?^[3]

You might protest that I am conflating two different meanings of “side”, that terminology sometimes clashes, and we just have to live with it. I am not so sure, though. The point of this exercise is to extrapolate from a situation (rectilinear figures) where the two notions mesh pretty well. If there was a new idea which captured everything we liked about the original but also applied to a broader category of shapes, then that would have an overwhelming claim to being the unique right answer. But if all our attempts at generalisation involve sacrificing desirable aspects of the original, then perhaps there is no single correct generalisation. There are different choices, with different trade-offs, which might be suitable in different contexts (and if we’re in a situation where more than one are in play, then they could helpfully be given different names).

Here’s a another variation: a Weierstrass tombstone created by replacing the top edge of a square by a section of Weiestress function, an infinitely wiggly line which isn’t smooth anywhere.

Here (and spot the typo) is what my Twitter friends made of this – although fewer ventured an opinion:

Notwithstanding the scepticism of my Twitter followers, I’ll explain in a minute why I don’t think it’s silly to see this as having four sides (one of which is non-smooth). On the other hand, if you prefer your sides smooth, then you again have a choice between seeing it has having infinitely many sides (three of which have length 1, and the rest having length 0), or having 3 sides plus a stretch of definitely-not-a-side boundary.

For one side: a single smooth curve.
Against one side: the same side on opposite sides.

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Off on a tangent 3: sides versus edges

How many sides does a square have? Four. How many edges does it have? Four. So are edges and sides the same thing? Not necessarily. Here are two configurations which are – at least arguably – each four-sided but have 5 and 3 edges respectively:

Usually, I would say, an “edge” is a topological object, in that its function, not its shape, is what matters. Think of the London tube map. If you asked how many edges there are in that network, there’s no merit in totting up straight or smooth sections. It’s connections between stations (or vertices) that count.

As already mentioned, it’s common to think of a polygon as a very simple sort of network called a closed polygonal chain: a string of vertices (in this case the corners of the polygon), linked with edges, in such a way that every vertex lies on exactly two edges, and the whole thing forms a single loop. In this situation edges and sides coincide, as do vertices and corners. But in general you can break this concurrence, as in the two small networks above.

If you want to think about things network-theoretically, but the vertices aren’t clearly marked, then you have to guess where they are. With a polygon this is easy – the vertices are at the corners – which is why switching between geometric and topological approaches comes so naturally. But with other shapes, such as either of the tombstones above, it may not be so obvious. Nevertheless, in each case if you were told there were vertices in there somewhere, and were asked to locate them, I think it would be sensible^[4] to guess that there are four, namely the corners of the original square, and that the top edge has for some reason been represented as a non-straight line. And if we do want to think about things that way, with each of the tombstones having 4 edges, then it might seem peverse (although logically coherent!) to insist that they have some other number of sides (especially as the top side is – despite its own geometry – clearly “on one side” of the figure). In fact, rather than guessing, one of my Twitter correspondents asked me “Have both top vertices been removed?”, a question which only makes sense from a network-theoretic perspective.

Where does this leave the circle? The trouble is that no point on the circle has a better claim to be vertex than any other. So although it is tempting (and again coherent) to view a circle as a network with one edge, if we are going to insert vertices, there is no obvious reason to prefer one to any other number.

Could we view it as a network with no vertices at all, a sort of Tube line with no stations? The usual mathematical conception of a network would not allow that, but that shouldn’t deter us too much. This suggests a purely topological approach. The trouble is that from that point of view, while a circle may be a sort of network with no vertices and one edge, so too is a square (if that’s how the Tube line happens to be laid out). In topology, a square is a circle. (This is not a paradox, it is simply saying that the boundary is a single loop, whose shape doesn’t matter.) So while this sort of network has “one edge”, obviously a square does not have “one side”, so the relationship between sides (geometric) and edges (topological) has again broken down, just as it does in the two little networks pictured above. So this appoach doesn’t take us very far forward.

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No Sides?

It might seem paradoxical to argue that a circle (or any shape) has “no sides”. But the argument for the defence is straightforward. We return to definition 2:

2. A straight line forming part of the boundary of a plane figure.

This is a simple, easily understandable phrase that perfectly captures the sides of a square. We have failed to find a satisfactory generalisation of this to curved figures, so the best thing to do is stick with the original. And a circle doesn’t have any.

For no sides: true, according to a sensible notion of “side”.
Against no sides: sounds like a Zen koan.

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None of the Above?

Recall the definition supplied by Google:

3. Each of the lines forming the boundary of a plane rectilinear figure.

Attempting to apply this to a circle – a non-rectilinear figure – produces nothing. The question is as meaningless as “How many sides does Monday have?”

Since Definition 3 is the most official (the only one in this post not made up by me or my children), doesn’t that make “None of the Above” categorically the right answer? Maybe. On the other hand: when someone poses us a question, the principle of charity perhaps requires us to assume that it is meaningful unless we can firmly establish otherwise, and definitions 1, 2, 4 and other variants make that possible. Further, definition 3 is linguistic rather than formally mathematical, and is therefore descriptive rather than prescriptive, so we should not be hidebound by it.

For none of the above: semantic malfunction.
Against none of the above: dialogic charity.

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The Right Answer

What prompted me to write this post? Like countless primary school students, my five-year-old twin sons – the primary school students mentioned at the start – were recently asked this question in their homework. One plumped for “1” and the other “0”, and I have tried to capture and expand on their reasoning above. I think both answers are wholly defensible – and neither is definitively right.

So, what should you do if you’re asked the question: How many sides does a circle have? In my opinion, the optimal response is to approach the mathematician in your life to write a 3000 word treatise on the topic, which you can then print out and triumphantly deliver to your unfortunate teacher. But failing that, the best approach is to follow the example of Socrates and respond to the question with a counter-question: What do you mean by “side”?

When all’s said and done, counting up to zero, or to one, or refusing to answer the question, tells us virtually nothing about the geometry of circles. But there is a lot to be gained by teasing apart familiar notions, dropping or adding extra conditions, challenging our intution by moving from one context to a slightly different one, and trying to write down precisely what we mean by a particular term in a particular setting. That’s what real mathematics is all about.

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Footnotes

[1] You could do something else: e.g. pick a starting point P on the circle, from which to measure distance around the circumference. Then declare that the points a rational distance from P are corners and rest are sides. This has effect of yielding a countably infinite number of corners and an uncountably infinite number of sides. Or one could stipulate the opposite. This might be a convenient fit for the polygonal limit approach to circles, but I’d struggle to agree that it’s easy or obvious enough to be considered “the right answer”.

[2] One of my Twitter correspondents was worried about how smooth the curve is. This tombstone is continuously differentiable but not twice so. It would certainly be interesting if many people thought this was a critical matter, and this could probably be tested with an infinitely smooth tombstone built from something like this, although I haven’t thought through the details.

[3] We might try to formalise this as follows: in a square (or any polygon), a side has the property that starting from any position in the interior, you can cut the shape straight through your location, so that your chosen side is firmly on one side of the cut. That doesn’t work for the two-sided tombstone: any cut will always sever the long side. We could weaken this by saying that to count as a side, there has to be at least one way of slicing through the shape so that the side is on one side of the cut. That would allow us to say that the tombstone has four sides (even though the curved section is not on one side of points in the upper region). For the circle though, its supposed one side is never on one side (so would be ruled out), and only the straight section of a semicircle would count as a side.

[4] It’s not easy to come up with a rigorous justification that works for both tombstones, but I am thinking more informally in terms of Schelling Points: that is to say locations which stand out as being special for reasons which may not be easy to predict in advance.

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Acknowledgements

Thanks to everyone who participated in or retweeted my polls, or discussed this with me on Twitter.