On the area discrepancy of triangulations of squares and trapezoids (Q551228)

Summary: In [Am. Math. Mon. 77, 161--164 (1970; Zbl 0187.19701)] \textit{P. Monsky} showed that a square cannot be triangulated into an odd number of triangles of equal areas; further, in [Discrete Math. 85, No.~3, 281--294 (1990; Zbl 0736.05028)] \textit{E. A. Kasimatis} and \textit{S. K. Stein} proved that the trapezoid \(T(\alpha)\) whose vertices have the coordinates \((0,0)\), \((0,1)\), \((1,0)\), and \((\alpha,1)\) cannot be triangulated into any number of triangles of equal areas if \(\alpha>0\) 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 \(T_{n_i}\) of the unit square into \(n_i\) triangles, \(n_i\) odd, so that the difference between the smallest and the largest area in \(T_{n_i}\) is \(O\left(\frac 1{n_i^3}\right)\). We then prove that for an arbitrarily fast-growing function \(f : \mathbb N \rightarrow \mathbb N\), there exists a transcendental number \(\alpha > 0\) and a sequence of triangulations \(T_{n_i}\) of the trapezoid \(T (\alpha )\) into \(n_i\) triangles, so that the difference between the smallest and the largest area in \(T_{n_i}\) is \(O\left(\frac 1{f(n_i)}\right)\)