Constructing equidissections for certain classes of trapezoids

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 spectrum S 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) . Key words: equidissection, spectrum, principal spectrum

• Type: Journal
• Year: 2008
• Where: DM
• Authors: Charles H. Jepsen, Paul Monsky
• Keywords: DM 2008, Equal Areas, Integer, Parallel Sides