Tilings of the plane with unit area triangles of bounded diameter

TL;DR

This paper demonstrates the existence of plane tilings using triangles with specific constraints on area and perimeter, addressing a problem posed by Nandakumar.

Contribution

It proves the existence of tilings with noncongruent triangles of equal area and bounded diameter, and also with equal perimeter and bounded area, solving Nandakumar's problem.

Findings

Existence of tilings with noncongruent triangles of equal area and bounded diameter

Existence of tilings with triangles of equal perimeter and bounded area

Addresses a longstanding problem by Nandakumar

Abstract

There exist tilings of the plane with pairwise noncongruent triangles of equal area and bounded perimeter. Analogously, there exist tilings with triangles of equal perimeter, the areas of which are bounded from below by a positive constant. This solves a problem of Nandakumar.