An aperiodic monotile, also somewhat humorously known as an einstein (where "einstein" means "one stone", perhaps generalizable to "one tile," in German), is a single tile that admits a tiling of a Euclidean space but no periodic tiling. The problem of finding such a tile in the plane was open for a long time. It was solved by Smith et al. (2023), who found a simple polykite known as a hat that is an aperiodic monotile.
In addition to the hat, Smith et al. (2023) also found a continuum of aperiodic polygonal monotiles. Every tiling by hats uses both handednesses. A spectre tile, introduced in the follow-up paper by Smith et al. (2024), admits only nonperiodic tilings of one handedness, even when reflections are allowed. This solves the strict form of the vampire einstein problem.
A monotile is called strongly aperiodic if it admits a tiling but no admitted tiling has a symmetry group containing an element of infinite order. Some authors instead reserve "strongly aperiodic" for the stricter condition that every admitted tiling has a trivial symmetry group, calling the finite-symmetry case mildly aperiodic (Tsiokos 2026).
Tsiokos (2026) constructed a three-dimensional monotile called Chair44 that is strongly aperiodic in the first sense. It is a polyhedron with rational vertex coordinates that is homeomorphic to a closed three-dimensional ball, formed from a seven-cube chair whose 24 exposed unit-square panels carry small square-pyramid features. Tsiokos proved that congruent copies, with reflections allowed, tile three-dimensional Euclidean space, but the symmetry group of every such tiling is finite (Pegg 2026, Tsiokos 2026).
Pegg, E. "Einstein Problem Solved (Aperiodic Monotile Discovery)."
Mar 21, 2023.