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(1n3i). 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(1f(ni))