Noncongruent equidissections of the plane

TL;DR

This paper explores tiling the plane with non-congruent triangles of equal area, demonstrating a tiling where perimeters are uniformly bounded, and discusses related variants of the problem.

Contribution

It proves the existence of a plane tiling with non-congruent equal-area triangles having bounded perimeters, advancing understanding of geometric tilings.

Findings

Existence of a tiling with non-congruent equal-area triangles and bounded perimeter

Several variants of the tiling problem are addressed and partially answered

Provides new insights into geometric tilings with area and perimeter constraints

Abstract

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.