Rolling with Reuleaux

8. The Bumpy Bike Path

Rolling with Reuleaux

A circular wheel isn't the only type of wheel that would ride smoothly over a flat, horizontal road.

Consider the problem of making a manhole cover with a shape that won't fall through an opening in the street. One answer is to use a circular lid that is slightly larger than the circular hole it covers. The lid can't slip through because it's wider than the hole, no matter which way you turn it. A circle has the same width no matter where you start on the edge to cross through the middle.

A standard circular manhole cover.

In contrast, an oval (or ellipse) is longer than it is wide. You can always find a way to slip an oval lid through an oval hole that is the same size or slightly smaller. That's also true of a square cover or a six-sided, or hexagonal, cover.

Amazingly, the circle isn't the only shape that would work safely as a manhole cover. Would that shape also make a smooth-riding wheel on a flat surface?

One possibility is the Reuleaux triangle, named after engineer Franz Reuleaux, who was a teacher in Berlin, Germany, more than a hundred years ago.

The Reuleaux triangle, unlike a standard equilateral triangle (dashed lines), has curved sides.

You might find an example of a Reuleaux triangle in your medicine cabinet. If you turn a bottle of NyQuil cough medicine or Pepto-Bismol stomach medicine upside down, the shape you see looks like a Reuleaux triangle. If you try rolling one of these bottles on its side, you'll find that it rolls nearly as smoothly as a round bottle.

The bottom of a bottle of NyQuil cough medicine has the shape of a triangle with curved sides, somewhat similar to a Reuleaux triangle.

TRY IT!

One way to draw a Reuleaux triangle is to start with an equilateral triangle, which has three sides of equal length. Place the pointed end of a pair of compasses at one corner of the triangle and stretch the arms until the pencil reaches another corner. Then draw an arc between two corners of the triangle. Draw two more arcs centered on the triangle's other corners.

Reuleaux's "curved triangle," as he called it, has a constant width—just like a circle. It can roll smoothly on a flat surface, like a circular wheel.

A pair of wheels shaped like Reuleaux triangles rolling along track in a display at the Exploratorium in San Francisco.

In fact, you can make a manhole cover or a wheel out of any regular polygon with an odd number of sides. Beginning with a five-sided shape (regular pentagon), for example, you can construct a rounded pentagonal shape with a constant width.

Reuleaux polygons with curved sides based on a regular pentagon (left) and regular heptagon (right).

Any wheel or other object with such a cross section would roll smoothly across your kitchen floor or down the street.

A manhole cover shaped like a Reuleaux triangle, found in San Francisco.

Imagine walking down the street and finding differently shaped manhole covers on every block!

See also "Manhole Cover Geometry," "Drilling a Square Hole," and "Reuleaux Triangle on the Street."

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