Constructing equidissections for certain classes of trapezoids

Abstract

We investigate equidissections of a trapezoid T (a), where the ratio of the lengths of two parallel sides is a. (An equidissection is a dissection into triangles of equal areas.) An integer n is in the spectrumS (T (a)) if T (a) admits an equidissection into n triangles. Suppose a is algebraic of degree 2 or 3, with each conjugate over Q having positive real part. We show that if n is large enough, n is in S (T (a)) iff n / (1 + a) is an algebraic integer. If, in addition, a is the larger root of a monic quadratic polynomial with integer coefficients, we give a complete description of S (T (a)). © 2007 Elsevier B.V. All rights reserved.