The Reuleaux Stones, by Jorge Buescu (Lisbon, Portugal)

A classic question in Microsoft job interviews in the 1990s was the following: why are manhole covers (or sewer lids) round?

The answer supposedly tells a lot about the way of thinking of the applicant. Of course the easiest way of transporting a heavy sewer lid is to make it roll on its side, and this would be extremely difficult were that lid to be oval or square. But this is not the real point: just looking at the streets of your hometown reveals that there are lots of square or rectangular lids for basic services, from electricity to grids for water drainage.

So, why are sewer lids round?

Here enters geometry! For understandable reasons, a basic requirement of sewer lids is that they seal hermetically whatever they’re covering. The way to do it is to build a small edge around the hole over which the lid will fit.

Suppose now that the lid is rectangular, square, hexagonal or oval. Whoever handles it will turn it around a bit, changing its orientation. However, any of these shapes is prone to a very unpleasant accident: a different orientation makes it possible for the lid to fall into the sewer, leaving as a consequence the very difficult (and unpleasant) job of lid-rescuing.

There is of course a simple way to avoid the “sewer lid accident”: using only circular lids. Opposite to the shapes mentioned above, the circle has constant diameter; whatever its orientation, its width (maximum distance between two different points) is constant. So there is no way that a circular lid can fall through a circular opening of the same diameter (in fact, slightly smaller because of the inner edge): it will always become “stuck”.

This would be the “right” answer for a Microsoft interviewer. To a mathematician, however, it suggests a new question: is the circle the only curve with the property of having constant width?

Maybe it seems counterintuitive, but there exists an infinite number of curves with the “constant-width” property. The first one was introduced by the German mechanical engineer Franz Reuleaux (1829—1905), a professor in Berlin, in the context of machine engineering. Constructed from an equilateral triangle, it became known as the “Reuleaux triangle” (fig. 1) although it is obviously not a triangle in the mathematical sense (Reuleaux called it a “curved triangle”).

 Figur 1. The Reuleaux triangle.

 Here´s the construction of the Reuleaux triangle. We start off with an equilateral triangle with side L. With center at each vertex we sequentially trace an arc of circle of radius L joining the two opposite vertices. These three arcs of circle compose the boundary of the Reuleaux triangle.

Now, given an arbitrary point P at this boundary, let us determine the width at P – that is, the maximum distance between P and any other point of this curve. Since P is on an arc of circle of radius L centered at the farthest vertex, it follows that the width at P is L. But P is an arbitrary point on the Reuleaux triangle. Thus the Reuleaux triangle has constant width L. 

It is easy to see that it is the constant-width property which is at stake when trying to avoid the sewer-lid accident. In fact, some cities use Reuleaux-shaped sewer lids; see fig. 2.

 
Fig. 2: Reulaeux-shaped sewer lids. Mathematically approved!

 More generally, we may apply Reuleaux’s construction to any odd-sided regular polygon, obtaining the so-called Reuleaux pentagons, Reuleaux heptagons and so on (see fig. 3) – an infinite family of constant-width curves. What about if we start with even-sided polygons? Well, in that case we still end up with a constant-width curve – but in the even-side case we always end up with a circle, so there’s nothing new to learn here.

 

Fig. 3: The first four Reuleaux polygons.

 Reuleaux polygons are much more than a simple curiosity. British people touch them literally every day: several coins in the British currency are Reuleaux heptagons (fig. 4). Given the wish for aesthetic originality in designing a non-round coin, why choose a Reuleaux shape? Because the constant-width property ensures that it will always roll smoothly, not becoming stuck for instance in vending machines.

 

Fig. 4: 20 e 50 pence UK coins.

 Reulaux devised his triangle in connection with machine optimization (remember, he was a mechanical engineer). He would probably be very happy to see his mathematical discovery applied industrially in the Wankel internal combustion engines, adopted in the 50s and 60s by many car manufacturers and still used today by Mazda (fig. 5).

 Fig. 5: A Reuleaux triangle, fundamental part in the Wankel rotation engine.

 What makes this mechanism tick? Like a circle, a Reuleaux triangle fits perfectly to a square whose sides are equal to the width of the curve, whatever the rotation imprinted. In fact, the Reuleaux triangle, when rotated without slipping describes an enveloping path which is “almost” a square – slightly rounded on the edges (fig. 6). This property allows for the development of rotary Reuleaux drills which, under rotation, drill an “almost” square hole! See e.g. http://www.youtube.com/watch?v=L5AzbDJ7KYI&feature=related.

 

Fig. 6: A rotating Reuleaux triangle describes “almost” a square (picture from Stan Wagon's book "Mathematica in Action", Springer-Verlag).

And the story does not end here. In 2009 Barry Cox, from the University of Wollongong (Australia) , and Stan Wagon, from Macalester College (USA) explored these geometrical ideas to solve the problem of drilling exact square holes. Their paper Mechanical circle-squaring (Coll. Math. Jour. 40.4 (2009): 238-247) they show how, starting from an isosceles right triangle, we may construct an adapted Reuleaux-Cox-Wagon “triangle” which under rotation drills holes which are exactly square. Stan Wagon himself has made a very elegant animation of this phenomenon, available online at http://www.youtube.com/watch?v=AWKW50d0oBM.

Maths is so rich that even sewage questions may prove a source of worthy problems! 

Jorge Buescu
August 2012