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Figures of Constant Width
© Copyright 2003, Jim Loy
At first thought, the circle would seem to be
the only two-dimensional figure of constant width. And as such, it can be used
as a wheel, without your vehicle bobbing up and down. But, it turns out that
there are infinitely many figures of constant width (depending on how you
define width). On the left we see two wheels of constant width (and an
animation on the right), and they too keep our vehicle (the thin rectangle on
top of our wheels) from bobbing up and down as they roll. It rolls as smooth as
can be (the two wheels don't have to be in synch with each other). The axle
does bob up and down, as you can see in the animation, so this wheel is of
limited practical value.
This figure is made up of three circular arcs of 60 degrees. It is the
second simplest of these figures of constant width (the circle being the
simplest). Such figures can be made from many circular arcs (the easiest to
construct), or with no circular arcs at all. The above diagrams were drawn with
the program Cinderella (the second one
with the help of Jasc's Animation Shop).
The above figure is called a Reuleaux triangle, and is also the shape of
the rotor in the Wankel rotory engine.
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