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MatheMUSEments
Covering Up
By Ivars Peterson
Muse, July/August, p. 36.
Have you ever wondered why the cover of a manhole is nearly
always round? Why couldn't it be oval or square?
The usual 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 would slip through a slightly
smaller hole of the same shape. That's because the circle has a
constant width—it's the same width all the way around.
In contrast, an oval 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 six-sided, or hexagonal, cover.
Amazingly, the circle isn't the only shape that would work
safely as a manhole cover. Another possibility is the Reuleaux
triangle, named after engineer Franz Reuleaux, who was a teacher
in Berlin, Germany, more than a hundred years ago. An example of
a Reuleaux triangle can be found in your medicine cabinet. If you
turn a bottle of NyQuil or Pepto-Bismol upside down, the shape
you see looks like a Reuleaux triangle.
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.
This "curved triangle," as Reuleaux called it, has a
constant width--just like a circle. It would certainly work as a
manhole cover.
In fact, you can make a manhole cover out of any shape with an
odd number of sides. Beginning with a five-sided shape called a
pentagon, for example, you can construct a rounded pentagonal
shape that has a constant width.
Imagine walking down the street and finding differently shaped
manhole covers on every block!
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