Drilling a Hexagonal Hole
By using a suitable roller restricted to rotate inside a hexagon, one can force a point on the roller to trace out an exact hexagonal path. This construction is due to the authors, inspired by a 1939 construction for a perfect square-hole drill. In this case, though, the roller does not touch all six sides of the hexagon, but it always touches sufficiently many sides to ensure rigidity within the ambient hexagon. As the roller rotates within the ambient hexagon, the center of the blue circle has a locus that is an exact hexagon. The red points are the points on the roller that touch the hexagon and enforce rigidity within it.
The construction of the roller proceeds as follows.
1. Draw a unit hexagon centered at .
2. Get the top arc using a circle of radius centered at (0,1).
3. Get the bottom arc using a circle of radius centered at (0,1). Let be the point of intersection of the bottom arc and the hexagon.
4. Get the northwest arc using center and radius . The northeast arc is similar.
5. Get the southwest arc using center and radius . The northeast arc is similar.
6. Use the intersection of the northwest and southwest arcs to define the upper termination of those arcs; the east side is similar.
As increases from 0 to the entire roller is rotated clockwise through and then translated leftward so as to place on the hexagon. For the first 0.049 radians of this rotation there are only four points of contact: the upper and lower points guarantee that there is no up-down motion, and the two points on the southwest and southeast sides guarantee rigidity in the left-right direction. After this critical value of there are five points of contact. As the rotation continues, symmetry governs the patterns and there are always four or five points of contact, sufficient to ensure rigidity.