Nu-configurations in tiling the square

Abstract

In order to tile the unit square with rational triangles, at least four triangles are needed. There are four candidate configurations: one is conjectured not to exist; two others are dealt with elsewhere; the fourth is the "nu-configuration," corresponding to rational points on a quartic surface in affine 3-space. This surface is examined via a pencil of elliptic curves. One rank-3 curve is treated in detail, and rational points are given on 772 curves of the pencil. Within the range of the search there are roughly equal numbers of odd and even rank and those of rank 2 or more seem to be at least 0.45 times as numerous as those of rank 0. Symmetrical solutions correspond to rational points on a curve of rank 1, which exhibits an almost periodic behavior.