Aperiodic Monotile Problem
A single tile can force nonperiodic tilings of the plane.
The aperiodic monotile problem asked whether one shape alone could tile the plane only nonperiodically. The 2023 'hat' and related monotiles gave a concrete answer to a long-running tiling question.
Formula
A single tile (monotile or einstein) that can tile the plane but only aperiodically — proved to exist in 2023.
The Smith–Myers–Kaplan–Goodman-Strauss hat belongs to a continuous family parameterized by two edge lengths.
No nonzero translation maps the tiling to itself — the defining property of an aperiodic tiling.
Summary
The aperiodic monotile problem asked whether one shape alone could tile the plane only nonperiodically. The 2023 'hat' and related monotiles gave a concrete answer to a long-running tiling question.
