Tilings with noncongruent triangles

TL;DR

This paper proves that the plane cannot be tiled with noncongruent triangles of equal area and perimeter, and that convex polygons with more than three sides cannot be finitely tiled with triangles without sharing full sides.

Contribution

It resolves a problem posed by R. Nandakumar by establishing non-existence results for specific triangle tilings and polygon tilings.

Findings

No tiling of the plane with pairwise noncongruent triangles of equal area and perimeter.

Convex polygons with more than three sides cannot be finitely tiled with triangles sharing no full sides.

Abstract

We solve a problem of R. Nandakumar by proving that there is no tiling of the plane with pairwise noncongruent triangles of equal area and equal perimeter. We also show that no convex polygon with more than three sides can be tiled with finitely many triangles such that no pair of them share a full side.