Riding with Reuleaux
Why is the cover of a manhole round? The usual answer is that a circular
lid, unlike a square or hexagonal cover, won't fall through the opening.
There's no way of orienting a round lid so that it fits through a hole
of the same geometry but slightly smaller size.
The circle works because it has a constant width. This width is defined
as the distance between a pair of parallel lines touching the curve on
opposite sides. For a circle, the width is simply the circle's diameter.
That's why wheels and cylindrical rollers produce a smooth ride on a flat
surface.
Unlike a circle, an ellipse doesn't have the same width in all directions.
So an elliptical lid could easily fall through an elliptical hole, and
an object riding on elliptical rather than cylindrical rollers would jiggle
up and down.
However, the circle isn't the only curve of constant width. There is
actually an infinite number of such curves, any one of which could form
a manhole lid or the cross section of a roller that gives as smooth a ride
as a cylinder.
Reuleaux triangle. 
The simplest such curve is known as the Reuleaux triangle, named after
engineer Franz Reuleaux, who taught in Berlin during the late nineteenth
century. One simple way to generate this figure is to start with an equilateral
triangle, then 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. This shape,
with rounded corners, may be familiar as the cross section of a bottle
of NyQuil or PeptoBismol. Its most prominent and successful application
may well be in the Wankel rotary internal combustion engine, which powers
several types of cars manufactured by Mazda. The engine features a curved,
triangular rotor turning in a specially shaped housing.
Like a circle, a Reuleaux triangle fits snugly inside a square
having sides equal to the curve's width no matter which way the triangle
is turned. Indeed, the rounded triangle can rotate freely inside the square
without ever having any room to spare.

Interestingly, as it rotates, the curved figure traces a path that eventually
covers just about every part of the square (except for a little rounding
at the corners). This property is the basis for an ingenious rotary drill
that, constrained by a special guide plate, bores square holes.
Reuleaux curves based on
the pentagon and the heptagon. 
It's possible to construct a curve of constant width not only from
an equilateral triangle but also from any polygon with an odd number of
sides. Thus, one can readily obtain a curved pentagon, heptagon, and so
on. Some coins have a rounded heptagonal shape that allows their use in
slot machines designed for ordinary coins. Drills shaped like curved heptagons
produce hexagonal holes.
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. So there's an unlimited number of curves of constant width, and the
Reuleaux triangle happens to be the family member of least area.
Why can't Reuleaux polygons be used in place of wheels? The trouble
is that these polygons don't have a fixed center of rotation. The hub of
a circular wheel, for example, stays a fixed height above the ground, allowing
smooth, horizontal motion. In contrast, the center of, say, a Reuleaux
triangle wobbles as the curve rotates. That doesn't matter for rollers
laid down on a surface to ease the passage of a heavy load, but it does
matter if the roller or wheel has a fixed axis. That's also why the drill
for cutting square holes requires a special "floating" chuck to hold the
drill. 
Copyright © 1996 by Ivars Peterson. 