Understanding when and why aperiodic monotiles occur

Ascertain the conditions under which a single prototile (monotile) admits only non-periodic tilings of the Euclidean plane, and identify principled methods or search spaces for locating additional aperiodic monotiles.

Background

Following the discovery of the hat and related shapes, the paper highlights the paucity of theory explaining the mechanisms that force aperiodicity in single-tile systems. Despite isolated examples, there is no general framework predicting where aperiodicity arises.

The author explicitly states the lack of knowledge regarding both the causes of aperiodicity and effective strategies to find more examples.