## August 14, 2010

### Manhole Cover Geometry

Why is the cover of a manhole nearly always round? Why isn't it oval or square?

Manhole covers are nearly always circular, but they may feature distinctive geometric designs, like the ring-and-spoke pattern shown here..

One answer is that a circular lid, unlike a square or an oval cover, won't fall through the opening. There's no way to position a round lid so that it can slip through a slightly smaller hole of the same shape. That's because a circle has a constant diameter. It has the same width all the way around.

In contrast, an oval or an ellipse is longer than it is wide. You can always find a way to slip an oval lid through a hole of the same shape. That's also true of a square or a hexagonal cover.

Plus signs add an arithmetical touch to this manhole cover in Pittsburgh.

A 12-sided (dodecagonal) manhole cover shouldn't work, but if the hole is sufficiently smaller than the lid, the cover's shape may be close enough to a circle to work effectively.

This 12-sided manhole cover near the U.S. Capitol features a triangular grid and a curious pattern of holes.

Amazingly, the circle isn't the only shape that would work safely as a manhole cover. In fact, any shape of constant width would do, and there are infinitely many such shapes. The simplest example is the Reuleaux triangle, named after distinguished mechanical engineer Franz Reuleaux (1829–1905), who was a teacher in Berlin more than 100 years ago.

A Reuleaux triangle (solid lines), based on an equilateral triangle (dashed lines) and set inside a square, is a geometric shape that has a constant width.

One way to draw a Reuleaux triangle is to start with an equilateral triangle, which has three sides of equal length. Draw three arcs of circles, with each arc having as its center one of the triangle's corners and as its endpoints the other two corners. The resulting “curved triangle," as Reuleaux termed it, has a constant width equal to the length of the interior triangle's side.

I haven't yet seen a manhole cover in the shape of a Reuleaux triangle, but I have heard that they exist.

One can construct a curve of constant width not only from an equilateral triangle but also from any polygon with an odd number of sides. Thus, you can readily obtain a curved pentagon, heptagon, and so on.

A Reuleaux pentagon and a Reuleaux heptagon (outer boundaries) are both curves of constant width.

Some coins have a rounded heptagonal shape that allows their use in slot machines designed for ordinary coins.

The Reuleaux curves described so far have corners—points where two sides meet at an angle. However, curves of constant width having rounded corners can be readily constructed from the angular forms.

Moreover, a curve of constant width need not be symmetrical or even consist of circular arcs. Therefore, an unlimited number of curves of constant width are possible, and the Reuleaux triangle happens to be the family member of least area.

Hexagons adorn a circular manhole cover in New Orleans.

In principle, manhole cover designers have an infinite array of shapes at their disposal, but circular covers remain the norm.

Photos by I. Peterson