These are geometric shapes which have the property that no matter what direction you orient them, their height is the same. The most obvious example is the circle, but there are others.

The most famous set of other shapes of constant width are based on regular polygons. Take a regular polygon with an odd number of sides, and construct arcs between each pair of adjacent vertices, each one centered at the vertex most opposite the two vertices connected by it. The smallest version is based on the triangle, and each arc is centered at one vertex and extends between the other two.

These shapes possess two interesting properties:

  1. Prisms with these shapes as cross-sections can be used as rollers, but unlike round shapes, they don't have a constant center, so they can't really be used as wheels. They can, however, be used as a strange sort of cam to generate a rather bizarre motion as they rotate inside a square hole.
  2. They possess one of the useful properties of manhole covers: you cannot drop the cover into the hole.

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