A friend we’ll call John (because that’s his name) just pointed me at a very interesting topic on the interweb.

John directed me to an article titled Mathematicians Discover New Shapes to Solve Decades-Old Geometry Problem on the Quanta Magazine website.

The 2-dimensional shape with constant width that pops into most of our minds is the circle. In this case we also refer to its width as it’s diameter. As we discover in this column, however, there are other shapes of constant width, perhaps the best known being the Reuleaux triangle.

What we mean by “constant width” is that if we sandwich these shapes between two parallel lines, they can roll smoothly along while constantly maintaining contact with both lines.

Circle and Reuleaux triangle (Click image to see a larger version -- Source: Max Maxfield)

A classic example of a Reuleaux triangle is presented in the form of the area of overlap in a three-circle Venn diagram. As per the Wikipedia, another way of describing all this is to say: “The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle. All points on a side of a Reuleaux triangle are equidistant from the opposite vertex in the equilateral triangle.” Well, that certainly rolls off the tongue, doesn’t it?

As an aside, I just ran across a related article, Drilling Square Holes with a Reuleaux Triangle, which is something I wouldn’t thought of myself in a thousand years. As part of this. I learned that a drill bit with a Reuleaux triangle cross section is known as a Harry Watt drill bit (this concept is nicely presented on the Wolfram Math World website).

The original article takes things further by presenting a video showing three non-spherical 3D objects with constant width rolling around between two horizontal planes.

The real point of all this is that a 2D Reuleaux triangle occupies less area than a circle of the same width. Similarly, a non-spherical 3D object of constant width occupies less volume than a sphere of the same width. Apparently, this facet of things (no pun intended) also applies to higher dimensions, the knowing of which I’m sure will prove really useful one day.

Now my head is spinning in multiple dimensions. How about you? Do you have any thoughts you'd care to share on this slippery shape shifting topic?