Tiling a circular disc with congruent pieces

TL;DR

This paper proves that monohedral tilings of a circular disc with up to three tiles must have rotational symmetry, providing new insights into the minimal number of tiles needed for such tilings.

Contribution

It establishes the first nontrivial bounds on the number of tiles in monohedral tilings of a circle with certain symmetry properties.

Findings

Monohedral tilings with up to 3 tiles have rotational symmetry.

First bounds on the minimum number of tiles with non-central tiles.

Progress towards a longstanding open problem.

Abstract

In this note we prove that any monohedral tiling of the closed circular unit disc with $k \leq 3$ topological discs as tiles has a $k$-fold rotational symmetry. This result yields the first nontrivial estimate about the minimum number of tiles in a monohedral tiling of the circular disc in which not all tiles contain the center, and the first step towards answering a question of Stein appearing in the problem book of Croft, Falconer and Guy in 1994.