Dissections of regular polygons into triangles of equal areas

Abstract

This paper answers the question, "If a regular polygon with n sides is dissected into m triangles of equal areas, must m be a multiple of n?" For n=3 the answer is "no," since a triangle can be cut into any positive integral number of triangles of equal areas. For n=4 the answer is again "no," since a square can be cut into two triangles of equal areas. However, Monsky showed that a square cannot be dissected into an odd number of triangles of equal areas. We show that if n is at least 5, then the answer is "yes." Our approach incorporates the techniques of Thomas, Monsky, and Mead, in particular, the use of Sperner's lemma and non-Archimedean valuations, but also makes use of affine transformations to distort a given regular polygon into one to which those techniques apply. © 1989 Springer-Verlag New York Inc.