# Incongruent equipartitions of the plane

## TL;DR

This paper investigates tilings of the plane with pairwise incongruent polygons of equal area, extending previous work by solving three new cases and demonstrating the existence of such tilings with triangles of unit area and bounded perimeter.

## Contribution

The paper provides new solutions to weaker versions of the incongruent equipartition problem, including the first construction of vertex-to-vertex tilings with incongruent triangles of equal area.

## Key findings

- Existence of vertex-to-vertex tilings with incongruent triangles of unit area.
- Construction of tilings with bounded perimeter for these triangles.
- Extension of previous results to new polygonal cases.

## Abstract

R. Nandakumar asked whether there is a tiling of the plane by pairwise incongruent triangles of equal area and equal perimeter. Recently a negative answer was given by Kupavskii, Pach and Tardos. Still one may ask for weaker versions of the problem, or for the analogue of this problem for quadrangles, pentagons, or hexagons. Several answers were given by the first author in a previous paper. Here we solve three further cases. In particular, our main result shows that there are vertex-to-vertex tilings by pairwise incongruent triangles of unit area and bounded perimeter.