by Scott Smith
published in The Mathematics Teacher, October 1993 (Volume 86 Number 7)
A bit that drills square holes ... it defies common sense. How can a revolving edge cut anything but a circular hole? Not only do such bits exist (as well as bits for pentagonal, hexagonal and octagonal holes), but they derive their shape from a simple geometric construction known as aReuleaux triangle (after Franz Reuleaux, 1829-1905).
To construct a Reuleaux triangle, start with an equilateral triangle of side s (Figure 1). With a radius equal to s and the center at one of the vertices, draw an arc connecting the other two vertices. Similarly, draw arcs connecting the endpoints of the other two sides. The three arcs form the Reuleaux triangle. One of its properties is thatof constant width, meaning the figure could be rotated completely around between two parallel lines separated by distance s.
It was with this property of constant width that the Reuleaux triangle was introduced in a sidebar of our geometry text (Moise and Downs, Teachers' Edition, p. 555). "This figure has constant width," I lectured, "just like a circle." Without thinking, I volunteered,"Imagine it as wheels on a cart." "What sort of cart?" "Why, a math cart, to carry my board compass and protractor," I replied, digging myself in deeper. This was the first of several impulsive misstatements I made about the Reuleaux triangle, only to admit after a little reflection that it wasn't so. Not in twenty years of teaching had my intuition failed me so completely.
The constant widthproperty can be used to transport loads, but not by using Reuleaux triangles as wheels. If several poles had congruent Reuleaux triangles as cross sections, bulky items could ride atop them (Figure 2). Movement would occur as poles were transferred from back to front, providing a moveable base of constant height.
But the Reuleaux triangle cannot be a wheel. The only conceivable point for the axle,at the triangle's centroid, is not the same distance from the Reuleaux triangle's "sides" (Figure 3). If the sides of the equilateral triangle are s, then
2 s sqrt(3)
(1) AP = - - sqrt(3) = ------- s » 0.577s,
3 2 3
sqrt(3) sqrt(3)PB = s - ------- s = s(1 - -------) » 0.423s.
Even if four Reuleaux triangle wheels were synchronized, the load would rise and fall continuously -- you'd need Dramamine to ride this cart!
"And since it has constant width, it would just fit inside a square whose sides are that width," I continued, trying toregain their attention. I carefully drew a square circumscribing the Reuleaux triangle (Figure 4). The triangle is normally tangent to two sides of the square with two vertices touching the square directly opposite the points of tangency (why?), as in Figure 4a. The exception is Figure 4b, where the Reuleaux triangle has one point of tangency and all three vertices on the square (one directlyopposite that point of tangency).
"If the Reuleaux triangle just fits inside the square, no matter what position it's in, couldn't it rotate around the inside of the square?" They needed convincing -- a model would have to be built. "But if it did rotate around the inside, doesn't that mean that a sharp Reuleaux triangle could carve out a square as it rotated?" I had them. "Drill a square hole?", onecountered. "No way!"
That night I cut a four inch Reuleaux triangle from a manila folder to take to class the next day. With a lot of effort, I was able to show the triangle rotate around the inside of a four inch square. "And if this was metal at the end of a rotating shaft, it would cut out a square", I continued, racking up two more falsehoods. Firstly, it was implied that the center of...