msb@lsuc.UUCP (Mark Brader) (04/15/85)
Gregory J.E. Rawlins (watdaisy!gjerawlins) posted a reference to Martin Gardner's Scientific American column for February 1963. So I looked it up. Here's the essence of what Gardner said: The properties of Reuleaux triangles* were discovered by Franz Reuleaux (1829-1905). He was an engineer, mathematician, and teacher, from Berlin. He wrote about them in his 1876 book The Kinematics of Machinery (Macmillan). The drill for square holes was invented by Harry James Watts, an English engineer then living in the US, in 1914. The bit is based on a Reuleaux triangle in the same way that a conventional bit is based on a circle -- it has flutes cut into the basic shape to let the debris out. Since the Reuleaux triangle has 120-degree angles, the holes ARE NOT perfectly square, but have slightly rounded corners. Gardner only gives a diagram and I haven't done the math myself, but the figure posted by Tye Cowan and Marc Zorn (trwrb!zorn), that only the last 5% or so of each straight side is lost to the curve, appears to jibe with the diagram. Of course, for practical purposes this may be sufficient, or the corners could be filed. The bit does indeed have to wobble in the hole. It is restrained by a square template, and mounted in a special "free-floating chuck". The design was covered by three U.S. Patents, numbers 1241175-7, issued September 25, 1917. The Watts Brothers Tool Works, of Wilmerding, PA, still existed at the time of the article; in addition to the square-hole drill, they were making drills for pentagonal, hexagonal, and octagonal holes. Gardner gives a shape for a triangular-hole drill (a lens-shaped cross-section) but notes that it would be mechanically very difficult to drive it in the necessary fashion. *Since I'm posting this to net.math as well as net.puzzle, some people are coming into it cold, so I'll define a Reuleaux "triangle". It's the shape formed by taking an equilateral triangle and joining each pair of vertices by a circular arc centered on the opposite vertex. It is the second-simplest, after the circle, of the infinite family of "curves of constant breadth". That means you can use a cylinder with that cross-section as a roller to move a flat object over a flat surface and the object won't bob up and down (though the roller's center of gravity will). If this isn't obvious, try it. Mark Brader
msb@lsuc.UUCP (Mark Brader) (04/16/85)
I write:
> The bit does indeed have to wobble in the hole.
Just to clarify, what I meant was that it doesn't rotate around a fixed axis.
It will be in contact with all sides of the square at any particular time.
Mark Brader