Colossal Gardner, ch. 4

We now move into Plane Geometry and the concept of Curves of Constant Width. Actually, the main content of Gardner’s article is captured in the wiki entry. His starting point is that you can use Reuleaux triangles in the place of wheels in a roller-based system, and a platform placed on top of the rollers will remain level and steady as the rollers rotate across a flat surface. So, it’s not necessary to use circular wheels if you don’t want to. (If you want pictures, go to the wiki article.)

The more interesting application of the Reuleaux triangle was in the Watts Brothers Tool Works patent for a drill and chuck system capable of drilling square holes. They also manufacture drill bits for pentagonal and hexagonal holes that have sharper corners. Although the outer edge of a curve of constant width is in contact with its bounding space at all times (as with a square for the Reuleaux triangle), the center of rotation moves around all over the place. That means that for the Watts square-hole bit, a special chuck is required to trace out the correct rotation to ensure that the hole actually comes out square. You can see the patents (filed in 1917) at Google patents: US1241175, US1241176 and US1241177.

The puzzle this time is based on the Kakeya Needle problem. What is the smallest convex area in which a line segment of length 1 can be rotated 360 degrees? (A convex figure is one in which a straight line, joining any two of its points, lies entirely on the figure. Examples include circles and squares.) Check the wiki article on Kakeya sets for an illustration of the Kakeya Needle.


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