Aperiodic tiling with "Tile(1,1)". The tiles are colored according to their rotational orientation modulo 60 degrees.[1] (Smith, Myers, Kaplan, and Goodman-Strauss)
In plane geometry, the einstein problem asks about the existence of a single prototile that by itself forms an aperiodic set of prototiles; that is, a shape that can tessellate space but only in a nonperiodic way. Such a shape is called an einstein, a word playonein Stein, German for "one stone".[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. The strictest version of the problem was solved in 2023 (the initial discovery was in November 2022, with a preprint published in March 2023), pending peer review.
In 1988, Peter Schmitt discovered a single aperiodic prototile in three-dimensional Euclidean space. While no tiling by this prototile admits a translation as a symmetry, some have a screw symmetry. The screw operation involves a combination of a translation and a rotation through an irrational multiple of π, so no number of repeated operations ever yield a pure translation. This construction was subsequently extended by John Horton Conway and Ludwig Danzer to a convex aperiodic prototile, the Schmitt–Conway–Danzer tile. The presence of the screw symmetry resulted in a reevaluation of the requirements for non-periodicity.[4]Chaim Goodman-Strauss suggested that a tiling be considered strongly aperiodic if it admits no infinite cyclic groupofEuclidean motions as symmetries, and that only tile sets which enforce strong aperiodicity be called strongly aperiodic, while other sets are to be called weakly aperiodic.[5]
The Socolar–Taylor tile was proposed in 2010 as a solution to the einstein problem, but this tile is not a connected set.
In 1996, Petra Gummelt constructed a decorated decagonal tile and showed that when two kinds of overlaps between pairs of tiles are allowed, the tiles can cover the plane, but only non-periodically.[6] A tiling is usually understood to be a covering with no overlaps, and so the Gummelt tile is not considered an aperiodic prototile. 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.[7] This construction requires matching rules, rules that restrict the relative orientation of two tiles and that make reference to decorations drawn on the tiles, and these rules apply to pairs of nonadjacent tiles. Alternatively, an undecorated tile with no matching rules may be constructed, but the tile is not connected. The construction can be extended to a three-dimensional, connected tile with no matching rules, but this tile allows tilings that are periodic in one direction, and so it is only weakly aperiodic. Moreover, the tile is not simply connected.
One of the infinite family of Smith–Myers–Kaplan–Goodman-Strauss tiles. Yellow tiles are the reflected versions of the blue tiles.
In November 2022, hobbyist David Smith discovered a "hat"-shaped tile formed from eight copies of a 60°–90°–120°–90° kite, glued edge-to-edge, which seemed to only tile the plane aperiodically.[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.[9][10] Furthermore, the hat can be generalized to an infinite family of tiles with the same aperiodic property. Their proof awaits peer review and formal publication.[11][12]
Tile(1,1) from Smith, Myers, Kaplan & Goodmann-Strauss on the left. A spectre is obtained by modifying the edges of this polygon as in the middle and right example.
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.[13] Furthermore, the "spectre" tile is a "strictly chiral" aperiodic monotile: even if reflections are allowed, every tiling is non-periodic and uses only one chirality of the spectre. That is, there are no tilings of the plane that use both the spectre and its mirror image.
^Two tiles have the same color when they can be brought in coincidence by the combination of a translation together with a rotation by an even multiple of 30 degrees. Tiles of different colors can be brought in coincidence by a translation together with a rotation by an odd multiple of 30 degrees.
