Equidissections of Trapezoids

Abstract

Suppose a plane polygon is dissected into triangles of equal areas. What numbers of triangles are possible? Given a polygon K, a dissection of K into m triangles of equal areas is called an m-equidissection. The spectrum of a polygon K, denoted S(K), is the set of integers m such that K has an m-equidissection. If S(K) consists of all multiples of a single integer m, we say that S(K) is principal and write S(K)= . Monsky [2] showed that if K is a square, then S(K)= . Kasimatis [1] showed that if K is a regular n-gon with n 2 5, then S(K) = . A discussion of this problem for other polygons is contained in Chapter 5 of [4]. What can we say about the spectrum of a trapezoid? We may confine our attention to trapezoids with vertices (0, 0), (1, 0), (0, 1), (a, 1), a > O, since any trapezoid is affinely equivalent to such a trapezoid. Denote this trapezoid by T(a). The following are shown in [4] (pp. 121-122):