On the Area Discrepancy of Triangulations of Squares and Trapezoids

In 1970 P. Monsky showed that a square cannot be triangulated into an odd number of triangles of equal areas; further, in 1990 E. A. Kasimatis and S. K. Stein proved that the trapezoid T (α) whose vertices have the coordinates (0, 0), (0, 1), (1, 0), and (α, 1) cannot be triangulated into any number of triangles of equal areas if α > 0 is transcendental. In this paper we first establish a new asymptotic upper bound for the minimal difference between the smallest and the largest area in triangulations of a square into an odd number of triangles. More precisely, using some techniques from the theory of continued fractions, we construct a sequence of triangulations Tni of the unit square into ni triangles, ni odd, so that the difference between the smallest and the largest area in Tni is O ( 1 ni ) . We then prove that for an arbitrarily fast-growing function f : N→ N, there exists a transcendental number α > 0 and a sequence of triangulations Tni of the trapezoid T (α) into ni triangles, so that the difference between the smallest and the largest area in Tni is O ( 1 f(ni) ) .