Circular Reasoning

Mathman strikes again. Here’s a nifty picture I have borrowed from Cracked.com. The text, by Cracked user RainbowCrash, says, “Manhole covers are round so that they cannot possibly fall through their own holes. Any shape other than a circle would be able to fit through in at least one way.”

Manhole covers dirty secret [RainbowCrash / Cracked.com]
Manhole covers dirty secret [RainbowCrash / Cracked.com]

Curiously, RainbowCrash’s profile advertises, “I’m not stupid anymore!” So let’s help him out here. There are shapes, other than a circle, that meet this definition. The requirement, when generalized, is that it must be a curve of constant width.

Wolfram defines this as “Curves which, when rotated in a square, make contact with all four sides. Such curves are sometimes also known as rollers.” I am not a fan of this definition, but it does describe the implications quite nicely. A circle is well known for this, but so is a curvilinear triangle, formally known as a Reuleaux triangle. Not only can this shape described a manhole cover, San Franciso uses them for it!

Noncircular manhole cover in San Francisco [Marianna Zavodovskaya / Wikimedia Commons]
Noncircular manhole cover in San Francisco [Marianna Zavodovskaya / Wikimedia Commons]

Cover image by Tomwsulcer / Wikimedia Commons.