Figures with constant thickness

releaux In addition to parallel lines, there are also parallel curves, which always keep the same distance. The sides of a road are parallel, as are the shores of a canal, and the guides on which the coins slide in a slot-machine. If we want the coins not to rattle while they fall, we must make it so that they always touch the sides of the guide. Round coins are good for that, since they can slide on two parallel curves always touching the two sides.

One might think that circles are the only figures with this property, that is the only curves with a constant thickness. In fact, this is not the case: we can construct many other figures that always touch two parallel curves. The simplest is a sort of triangle with the sides formed by circular arches. sliding inside a guide with parallel sides. From this, cutting near the angles to make them sharp, we obtain the point of an (eccentric) drill to make square holes. One can find drills to make hexagonal holes on sale.
One can also make curves of constant thickness with various shapes. The English 50-pence coins are among those.

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