Monohedral Tilings of a Convex Disc with a Smooth Boundary

TL;DR

This paper characterizes all normal monohedral tilings of convex discs with smooth boundaries, extending previous results, and also explores partial results for non-normal tilings.

Contribution

It provides a comprehensive description of such tilings with up to three tiles, generalizing earlier work and including partial results for non-normal cases.

Findings

Complete classification of normal monohedral tilings with up to three tiles

Extension of previous tiling results to broader classes

Partial results for non-normal tilings

Abstract

In this paper we give a complete description about normal monohedral tilings of a convex disc with smooth boundary where we have at most three topological discs as tiles. This result is a far-reaching generalization of the results of Kurusa, L\'angi and V\'igh \cite{KLV2020}. Some further partial results are proved for non-normal tilings.