Abstract
To tile consists in assembling colored tiles on \({{\mathbb Z}^2}\) while respecting color matching. Tilings, the outcome of the tile process, can be seen as a computation model. In order to better understand the global structure of tilings, we introduce two topologies on tilings, one à la Cantor and another one à la Besicovitch. Our topologies are concerned with the whole set of tilings that can be generated by any tile set and are thus independent of a particular tile set. We study the properties of these two spaces and compare them. Finally, we introduce two infinite games on these spaces that are promising tools for the study of the structure of tilings.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Allauzen, C., Durand, B.: The Classical Decision Problem. In: Appendix A: “Tiling problems”, pp. 407–420. Springer, Heidelberg (1996)
Berger, R.: The undecidability of the domino problem. Memoirs of the American Mathematical Society 66, 1–72 (1966)
Cervelle, J., Durand, B.: Tilings: recursivity and regularity. Theoretical Computer Science 310(1-3), 469–477 (2004)
Culik II, K., Kari, J.: On aperiodic sets of Wang tiles. In: Foundations of Computer Science: Potential - Theory - Cognition, pp. 153–162 (1997)
Durand, B., Levin, L.A., Shen, A.: Complex tilings. In: Proceedings of the Symposium on Theory of Computing, pp. 732–739 (2001)
Durand, B.: Tilings and quasiperiodicity. Theoretical Computer Science 221(1-2), 61–75 (1999)
Durand, B.: De la logique aux pavages. Theoretical Computer Science 281(1-2), 311–324 (2002)
Gale, F., Stewart, F.M.: Infinite games with perfect information. Ann. Math. Studies 28, 245–266 (1953)
Hanf, W.P.: Non-recursive tilings of the plane. I. Journal of Symbolic Logic 39(2), 283–285 (1974)
Lafitte, G., Weiss, M.: Universal Tilings. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 367–380. Springer, Heidelberg (2007)
Martin, D.A.: 1975. Annals of Math. 102, 363–371 (1975)
Myers, D.: Non-recursive tilings of the plane. II. Journal of Symbolic Logic 39(2), 286–294 (1974)
Oxtoby, J.C.: Contribution to the theory of games, Vol. III. Ann. of Math. Studies 39, 159–163 (1957)
Robinson, R.: Undecidability and nonperiodicity for tilings of the plane. Inventiones Mathematicae 12, 177–209 (1971)
Serre, O.: Contribution á l’étude des jeux sur des graphes de processus á pile, PhD thesis, Université Paris VII (2005)
Wang, H.: Proving theorems by pattern recognition II. Bell System Technical Journal 40, 1–41 (1961)
Wang, H.: Dominoes and the \(\forall\exists\forall\)-case of the decision problem. In: Proceedings of the Symposium on Mathematical Theory of Automata, pp. 23–55 (1962)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lafitte, G., Weiss, M. (2008). A Topological Study of Tilings. In: Agrawal, M., Du, D., Duan, Z., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2008. Lecture Notes in Computer Science, vol 4978. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79228-4_33
Download citation
DOI: https://doi.org/10.1007/978-3-540-79228-4_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79227-7
Online ISBN: 978-3-540-79228-4
eBook Packages: Computer ScienceComputer Science (R0)