6 min read

Reinventing the wheel

In this extremely niche post I discuss a few different ways of constructing a circle. If you’re not interested in the accompanying fluff, skip straight to the definitions numbered below.


Pi day

When I was working as one of the tutors at AIMS-Cameroon, we were asked by a local secondary school to put a talk together on Pi day (every 14 March, of course) for their pupils. We were asked that ideally it should be about pi but really anything mathsy would do. The pupils were old enough to know that pi is the ratio of the circumference of a circle to its diameter, but to ensure everyone was on the same page we started with a quick demonstration:

Roll a wheel along a table until it’s gone all the way round, measure the distance rolled (circumference) in terms of the width of the coin (diameter). It’s 1… 2… 3… and a bit wheel-widths. Or pi. Or \(\pi\). This is true no matter what size the wheel.

Pi-unrolled-720

The talk continued about how \(\pi\) is transcendental, but can be approximated by this equation or approached in that limit and is useful here and pops up there and so on. Then we ran out of ideas and moved on along other mathsy tangents. I’m not sure whether we held the attention of many pupils but we1 got a round of applause at the end.

At no point in the talk did we actually bother to define a circle – we had limited time and everybody knows what a circle is anyway, right? Yet before we made this decision, I had made a load of notes about how circles can be defined or constructed in different ways. Essentially making notes on how to reinvent the wheel. To avoid these notes being not only pointless (angles) but also pointless (teleology), I wrote this blog. Hopefully it will strike a chord with someone.

Circles all round

Most people past a certain age should be able to define a circle. To draw a circle, or near-enough. To give examples of circular things. To discuss the various properties that circles have. At the very least they should be able to point out a circle and a non-circle. A circle is a shape that’s round, without corners. What about a spiral? OK, it’s a closed, round shape. What about a ring? OK, it’s a closed, round shape with one boundary. What about an oval? OK, it’s a closed, round shape with one boundary that’s the same all the way around. What about a sphere? OK, it’s

1. A flat, closed, round shape with one boundary that’s the same all the way around.

A flat, closed, round shape with one boundary that’s the same all the way around? Is that the standard definition? Well, no, but while it’s not rigourous or particularly elegant, it does capture the essential characteristics of a circle – two-dimensionality, roundness, uniformity.

By contrast, here’s more or less the standard definition:

2. The set of all points on a plane that are the same distance \(r\) from a fixed point \(P\) is a circle.

It expresses a circle as an infinite set of points satisfying a particular condition (a locus) and it is surely more elegant and rigourous than 1. But it doesn’t explicitly reveal much about the circle’s character, nor how to construct one. Instead, we can express a circle as a closed curve rather than a set of points:

3. Take a line segment of fixed length \(r\). If one endpoint \(Q\) is rotated around the other fixed endpoint \(P\) then the path traced by \(Q\) is a circle.

This essentially puts the set of points in 2 in the order that they are visited by \(Q\). If we state the rule of motion of \(Q\) differently then we get a different definition. For instance, consider the curvature of the path:

4. Take a point \(Q\) on a plane. If \(Q\) is moved in an initial direction and continues such that its curvature \(k\) is constant then the path traced by \(Q\) is a circle.

This is like driving a car with the wheels turned at a fixed angle. This time we don’t need to explicitly define the centre to get the the circle, though fixing the wheels is equivalent to fixing the curvature which is equivalent to fixing the radius, since \(k=\frac{1}{r}\).

Circles also exist as solutions to constrained optimisation problems. For instance, we can make any arbitrary loop (a simple closed curve) and into a circle like this:

5. Take a loop whose perimeter \(c\) is fixed. The shape that maximises the area is a circle.

Or like this:

6. Take a loop whose area \(a\) is fixed. The shape that minimises the perimeter is a circle.

Or like this:

7. Take a loop whose area or perimeter is fixed. The shape that minimises the maximum distance of the boundary from the centroid is a circle, and the shape that maximises the minimum distance of the boundary from the centroid is a circle.

Notice that all definitions above include a property that determines the circle’s size – either \(r\), \(k\), \(c\), or \(a\) – but only 2-4 determine the circle’s location, using \(P\) or the initial location and direction of \(Q\). To fix the location for 5-8 we could, for example, fix the centroid of the shape or the highest point on the boundary.

The last I can think of is:

8. The limit of a regular polygon as the number of sides or corners goes to infinity is a circle.

This basically says that an equilateral triangle is to a square, what a square is to a regular pentagon, what a regular pentagon is to a regular hexagon, what a regular hexagon is to a regular septagon, what a… is to a circle. Here, neither the size or location is defined so we simply have a geometric shape.

What’s the point?

Each of these constructions have different advantages. 2 let’s you know where not to stand when approaching an angry dog on a chain. 3 is why compasses work. 4 helps the football groundkeeper to mark out a centre circle. 5 can be used by a farmer with limited fencing to increase space for her herd. 6 is why pipes have circular cross-sections. 7 – pass. 8 is useful for representing circles in discrete space, for example with pixels.

Still here?

After a bit of searching I found two more locus-based definitions but with a slightly more complex condition than in 2:

9. Take two points \(A\) and \(B\). The locus of points \(Q\) such that the angle \(\angle AQB\) is a right-angle is a circle. (This is an inversion of Thales’ theorem)

And:

10. Take two points \(K\) and \(L\). The locus of points \(Q\) such that the ratio \(\overline{KQ}/\overline{LQ}\) is constant is a circle. (This is the Circle of Apollonius)

Presumably there are many more locus-based and non-locus-based ways to make circles. What have I missed?


Loosely related song and recent obsession:


  1. I say “we”. Though I helped to prepare the talk, it was given by three of my colleagues in French so I didn’t do much on the day except take a few photos.