Wang's conjecture states that if a set of tiles can tile the plane, then they can always be arranged to do so periodically (Wang 1961). The Conjecture was refuted when Berger (1966) showed that an aperiodic set of tiles existed. Berger used 20,426 tiles, but the number has subsequently been greatly reduced.

**References**

Adler, A. and Holroyd, F. C. ``Some Results on One-Dimensional Tilings.'' *Geom. Dedicata* **10**, 49-58, 1981.

Berger, R. ``The Undecidability of the Domino Problem.'' *Mem. Amer. Math. Soc. No.* **66**, 1-72, 1966.

Grünbaum, B. and Sheppard, G. C. *Tilings and Patterns.* New York: W. H. Freeman, 1986.

Hanf, W. ``Nonrecursive Tilings of the Plane. I.'' *J. Symbolic Logic* **39**, 283-285, 1974.

Mozes, S. ``Tilings, Substitution Systems, and Dynamical Systems Generated by Them.'' *J. Analyse Math.* **53**, 139-186, 1989.

Myers, D. ``Nonrecursive Tilings of the Plane. II.'' *J. Symbolic Logic* **39**, 286-294, 1974.

Robinson, R. M. ``Undecidability and Nonperiodicity for Tilings of the Plane.'' *Invent. Math.* **12**, 177-209, 1971.

Wang, H. *Bell Systems Tech. J.* **40**, 1-41, 1961.

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1999-05-26