Noncongruent Equidissections of the Plane
Nandakumar asked whether there is a tiling of the plane by pairwise non-congruent triangles of equal area and equal perimeter. Here a weaker result is obtained: there is a tiling of the plane by pairwise non-congruent triangles of equal area such that their perimeter is bounded by some common constant. Several variants of the problem are stated, some of them are answered.
The author expresses his gratitude to R. Nandakumar for providing several interesting problems. Particular thanks to one of the anonymous referees for providing the “no” in Table 1 and for other valuable remarks. Special thanks to Jens Schubert for helpful discussions on this topic during a pleasant summer weekend in Bochum.