Tilings with noncongruent triangles (Q1663799)

A family of (closed) triangles is said to form a tiling if together the triangles fill the whole plane with no gaps and no overlaps. A few years ago, R. Nandakumar proposed the following question ``Is it possible to tile the plane with pairwise noncongruent triangles of the same area and perimeter?'' The authors in this paper show that the answer to this question is false. Moreover, they also prove that any tiling of a convex polygon with at least four sides by finitely many triangles contains necessarily a pair of triangles that share a full side.