A mechanism for drilling square holes has to turn circular motion into square motion.
In one early attempt to create such a device, James Watts had the idea of rotating a Reuleaux triangle within a square. A Reuleaux triangle, named after mechanical engineer Franz Reuleaux (1829–1905), has the same width all the way around. Its shape is made from arcs of circles centered at the vertices of an equilateral triangle.
To make a Reuleaux triangle, draw three arcs of circles, with each arc having as its center one of an equilateral triangle's vertices and as its endpoints the other two vertices.
Like a circle, this rounded triangle fits snugly inside a square having sides equal to the curve's width no matter which way the triangle is turned. As it rotates, the curved figure traces a path that eventually covers nearly every part of the square. Watts started a company, Watts Brothers Tool Works in Wilmerding, Pa., to make square-hole drills based on this idea. The company is still in operation today.
However, the resulting drilled shape is not a perfect square. Its corners are slightly rounded.
Rotated inside a square, a Reuleaux triangle traces a curve that is almost a square.
Barry Cox (University of Wollongong) and Stan Wagon (Macalester College) have recently explored geometric solutions to the problem of drilling exact square holes. They describe their investigations in the article "Mechanical Circle-Squaring," published in the September College Mathematics Journal.
Cox and Wagon begin with a mechanical device originally presented in an anonymous 1939 article in the magazine Mechanical World. John Bryant and Chris Sangwin (University of Birmingham) revisited the design in their book How Round Is Your Circle? Where Engineering and Mathematics Meet (Princeton University Press, 2008) and built a physical model of the drill.
The geometric key is to use a variant of the classic Reuleaux triangle in which one vertex is rounded off. The starting point is an isosceles right triangle. In the completed construction, the vertex at the right angle traces out a small square when the entire figure rotates within a larger square.
In this variant of the classic Reuleaux triangle, the vertex C traces out the inner square (dashed lines) when the rotor, a curve of constant width, rotates so that it always lies within the outer square.
"If one places a cutting tool at point C . . . and turns the rotor so that it stays inside the large square, then C traces out an exact square and the cutting tool stays inside the larger square," Cox and Wagon write. "Thus the device can be viewed as a drill that drills an exact square hole, though we need to bring the construction into the third dimension to get a working model."
Using computer algebra tools, Cox and Wagon generalized the approach to obtain a roller for drilling a perfect hexagonal hole. In this case, the required shape is made up of six circular arcs.
A roller for drilling a perfect hexagonal hole is made up of six circular arcs, centered at O, A, X, O, Y, and F.
"Similar ideas work for the octagon, and it seems likely that they will extend to regular n-gons when n is even," Cox and Wagon conclude. "So the main unresolved problem is whether one can construct a device along these lines that will make a 3- or 5-sided hole."
Cox, B., and S. Wagon. 2009. Mechanical circle-squaring. College Mathematics Journal 40(September):238-247.
Peterson, I. 2003. Rolling with Reuleaux. MAA Online (Sept. 22).