Einstein problem

This article is about the geometric problem. For the logic puzzle, see Zebra Puzzle. For other uses, see Einstein (disambiguation).

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)

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 play on ein 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, after an initial discovery in 2022.

The einstein problem can be seen as a natural extension of the second part of Hilbert's eighteenth problem, which asks for a single polyhedron that tiles Euclidean 3-space, but such that no tessellation by this polyhedron is isohedral.[3] Such anisohedral tiles were found by Karl Reinhardt in 1928, but these anisohedral tiles all tile space periodically.

  1. ^ 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.
  2. ^ Klaassen, Bernhard (2022). "Forcing nonperiodic tilings with one tile using a seed". European Journal of Combinatorics. 100 (C): 103454. arXiv:2109.09384. doi:10.1016/j.ejc.2021.103454. S2CID 237571405.
  3. ^ Senechal, Marjorie (1996) [1995]. Quasicrystals and Geometry (corrected paperback ed.). Cambridge University Press. pp. 22–24. ISBN 0-521-57541-9.