Einstein problem

[2] Several variants of the problem, depending on the particular definitions of nonperiodicity and the specifications of what sets may qualify as tiles and what types of matching rules are permitted, were solved beginning in the 1990s.

In 1988, Peter Schmitt discovered a single aperiodic prototile in three-dimensional Euclidean space.

This construction was subsequently extended by John Horton Conway and Ludwig Danzer to a convex aperiodic prototile, the Schmitt–Conway–Danzer tile.

An aperiodic tile set in the Euclidean plane that consists of just one tile–the Socolar–Taylor tile–was proposed in early 2010 by Joshua Socolar and Joan Taylor.

[8] Smith recruited help from mathematicians Craig S. Kaplan, Joseph Samuel Myers, and Chaim Goodman-Strauss, and in March 2023 the group posted a preprint proving that the hat, when considered with its mirror image, forms an aperiodic prototile set.

[11] In May 2023 the same team (Smith, Myers, Kaplan, and Goodman-Strauss) posted a new preprint about a family of shapes, called "spectres" and related to the "hat", each of which can tile the plane using only rotations and translations.

Aperiodic tiling with "Tile(1,1)", a type of the Spectre tiles. The tiles are colored according to their rotational orientation modulo 60 degrees. [ 1 ] (Smith, Myers, Kaplan, and Goodman-Strauss)
The Socolar–Taylor tile was proposed in 2010 as a solution to the einstein problem, but this tile is not a connected set.