My first blog post was inspired by my visits to the Museum of Mathematics, and it appears that my second will be as well. Like the first post this one will be suitable for a general audience, but I’ll write a follow-up which gets a little more mathematically serious.

I started volunteering at MoMath as an “integrator” this past Sunday; the primary role of the integrator is to float around the museum helping people who seem confused by an exhibit. A secondary role is to operate the few exhibits which require staff supervision, and I was taught how to operate an exhibit called the “coaster roller”. The coaster roller works as follows. There is a small pit full of peculiar-looking objects and a sort of “raft” which sits on top of them (I’ll take a picture next time I visit the museum). The objects are about the size of basketballs, but they are not spherical. Here’s a picture of what they might look like, stolen from the internet:

A person (or several children) can sit in the raft and pull themselves along using ropes, rolling over the strange shapes at the bottom.

So what’s so special about the shape of these objects? The important property that they each possess is that they have *constant width*, meaning that if one of the objects fits in a vise when it is pointing in one direction then it fits in the same vise when it is pointing in any other direction. This is important because if the width varied then the distance between the raft and the floor would vary as it rolled along and the ride would get pretty bumpy!

The constant width property is possessed by a sphere but not by a cube: the width of a cube is larger when measured between opposite corners than when measured between opposite faces. If you think about it for a moment, you might find it hard to convince yourself that bodies of constant width which aren’t spheres can even exist! But in fact they exist in abundance; I will explain a procedure which allows you to construct infinitely many different ones.

To begin, note that given any plane curve of constant width one can obtain a corresponding surface of constant width by rotating the curve in three-dimensional space (you may have already noticed that each of the surfaces in the picture above is rotationally symmetric). Not all surfaces of constant width arise this way, but in any event this shows that it is enough to look at curves of constant width if we are happy just finding a few basic examples.

To construct a curve of constant width, begin by drawing an equilateral triangle with side length . Then draw the circle of radius centered at each of the three vertices of the triangle. The boundary of the intersection of the disks enclosed by these circles is then a curve of constant width . Can you figure out why? Here’s a picture, stolen from Wikipedia:

In fact, the same procedure works if you start with any regular polygon with an odd number of sides (thanks to Ben Levitt from MoMath who corrected my original claim that it works for any regular polygon). The curves of constant width obtained in this way are often called *Reuleaux polygons* for their discoverer (even though they aren’t technically polygons).

There are all sorts of interesting mathematical questions one can ask about bodies of constant width. Here are some useful facts:

- All curves of the same constant width have the same perimeter.
- Among all curves with a given constant width, the circle encloses the largest area. Similarly, among all surfaces with a given constant width, the sphere encloses the largest volume.
- Among all curves with a fixed constant width, the Reuleaux triangle encloses the smallest area.
- Among all surfaces with a fixed constant width, nobody knows which one encloses the smallest volume!

In my next blog post, I plan to discuss some of these facts in greater detail. If you want to read more in the meantime, I recommend The Enjoyment of Math by Rademacher and Toeplitz, two great 20th century mathematicians. Actually, I recommend that book even if you don’t care to read more about bodies of constant width!

rodkimball

Jan 30, 2013@ 20:21:52I just proved the notion that regular reuleaux “polygons” with the same diameter also have the same circumference. It’s surprisingly simple. Rod

pwsiegel

Jan 30, 2013@ 23:00:23In fact any two curves with the same constant width have the same circumference, even if they aren’t of Reuleaux type. I’m going to post a proof of this using Crofton’s formula soon; see if you can figure it out!

Ethan Bolker

Apr 08, 2014@ 17:32:48Very nice. I’m looking forward to following. You might want to edit this to change “vice” to “vise” (and delete this comment then if you like).

Ryan Sanders

May 18, 2014@ 16:06:01Thank you. This was very informative. I’ve never been to MoMath but I just recently stumbled across it and am definitely looking into a trip there this year. I’ll be including a link to your blog (with full credit to you) in an upcoming entry I’m doing on education and museums like this for my own blog. Once again, thank you, very well written.