Abstract
We construct the first known example of a strongly aperiodic set of tiles in the hyperbolic plane. Such a set of tiles does admit a tiling, but admits no tiling with an infinite cyclic symmetry. This can also be regarded as a “regular production system” [5] that does admit bi-infinite orbits, but admits no periodic orbits.
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References
Berger, R.: The undecidability of the Domino Problem. Mem. Am. Math. Soc. 66 (1966)
Block, J., Weinberger, S.: Aperiodic tilings, positive scalar curvature and amenability of spaces. J. Am. Math. Soc. 5, 907–918 (1992)
Epstein, D., et al.: Word processing in groups. Boston: Jones and Bartlett 1992
Goodman-Strauss, C.: Open questions in tilings. Preprint
Goodman-Strauss, C.: Regular production systems and triangle tilings. Preprint
Grünbaum, B., Shepherd, G.C.: Tilings and patterns. New York: W.H. Freeman and Co. 1987
Kari, J.: A small aperiodic set of Wang tiles. Discrete Math. 160, 259–264 (1996)
Margulis, G.A., Mozes, S.: Aperiodic tilings of the hyperbolic plane by convex polygons. Isr. J. Math. 107, 319–325 (1998)
Mozes, S.: Aperiodic tilings. Invent. Math. 128, 603–611 (1997)
Penrose, R.: Pentaplexity. Math. Intell. 2, 32–37 (1978)
Robinson, R.M.: Undecidability and nonperiodicity of tilings in the plane. Invent. Math. 12, 177–209 (1971)
Robinson, R.M.: Undecidable tiling problems in the hyperbolic plane. Invent. Math. 44, 259–264 (1978)
Wang, H.: Proving theorems by pattern recognition II. Bell System Tech. J. 40, 1–42 (1961)
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Goodman-Strauss, C. A strongly aperiodic set of tiles in the hyperbolic plane. Invent. math. 159, 119–132 (2005). https://doi.org/10.1007/s00222-004-0384-1
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DOI: https://doi.org/10.1007/s00222-004-0384-1