Cut The Knot!An interactive column using Java applets
by Alex Bogomolny
At the end of the August's column we arrived at Barbier's theorem as a surprising consequence of Count Buffon's experiment. This time around I wish to present another demonstration (found in [Yaglom]) that stems from a surprising and elegant application of algebraic concepts to geometry.
Barbier's theorem states that all shapes of constant width D have the same perimeter pD. The width of a convex figure in a certain direction is the distance between two supporting lines perpendicular to that direction. (A straight line is called supporting a convex figure if they have at least one common point and the figure lies on one side from the line. In any direction there are two supporting lines.) Shapes of constant width are convex figures that have the same width in any direction. The circle has this property. The Reuleaux triangle is the next simplest shape that does. Others may be constructed starting with equilateral (but not necessarily equiangular) stars, as demonstrated by the applet below.
The algebraic concept fundamental to the proof is Minkowski's addition of convex sets. With a fixed origin O, the sum of two shapes is the collection of all endpoints of vectors OA+OB, where A ranges over one set, B over the other. Several important properties of Minkowski's addition could be discerned with the help of the following applet that shows the sum of two ("Left" and "Right") polygons. (Polygons and their vertices are draggable, as is the origin.)
Properties of Minkowski's Addition
To complete the toolbox for a proof of Barbier's theorem we only need one additional fact about shapes of constant width:
We are now in a position to prove Barbier theorem. Let K be a shape of constant width D. Let L be obtained from K by central symmetry. L is also a shape of constant width with diameter D. The following can be claimed about the sum K+L:
If p(K) is the perimeter of K and P(L) is that of L, then
Alex Bogomolny has started and still maintains a popular Web site Interactive Mathematics Miscellany and Puzzles to which he brought more than 10 years of college instruction and, at least as much, programming experience. He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem. He can be reached at firstname.lastname@example.org
Copyright © 1996-2001 Alexander Bogomolny