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Arithmetic discrete planes are quasicrystals. (English) Zbl 1261.52013
Brlek, Srečko (ed.) et al., Discrete geometry for computer imagery. 15th IAPR international conference, DGCI 2009, Montréal, Canada, September 30 – October 2, 2009. Proceedings. Berlin: Springer (ISBN 978-3-642-04396-3/pbk). Lecture Notes in Computer Science 5810, 1-12 (2009).
Summary: Arithmetic discrete planes can be considered as liftings in the space of quasicrystals and tilings of the plane generated by a cut and project construction. We first give an overview of methods and properties that can be deduced from this viewpoint. Substitution rules are known to be an efficient construction process for tilings. We then introduce a substitution rule acting on discrete planes, which maps faces of unit cubes to unions of faces, and we discuss some applications to discrete geometry.
For the entire collection see [Zbl 1176.68004].

52C23 Quasicrystals and aperiodic tilings in discrete geometry
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
68R15 Combinatorics on words
Full Text: DOI
[1] Andres, É., Acharya, R., Sibata, C.: The Discrete Analytical Hyperplanes. Graph. Models Image Process. 59, 302–309 (1997) · doi:10.1006/gmip.1997.0427
[2] Arnoux, P., Ito, S.: Pisot substitutions and Rauzy fractals. Bull. Bel. Math. Soc. Simon Stevin 8, 181–207 (2001) · Zbl 1007.37001
[3] Arnoux, P., Berthé, V., Ito, S.: Discrete planes, \(\mathbb{Z}\)2-actions, Jacobi-Perron algorithm and substitutions. Ann. Inst. Fourier (Grenoble) 52, 1001–1045 (2002) · Zbl 1017.11006 · doi:10.5802/aif.1889
[4] Arnoux, P., Berthé, V., Fernique, T., Jamet, D.: Functional stepped surfaces, flips and generalized substitutions. Theoret. Comput. Sci. 380, 251–267 (2007) · Zbl 1119.68136 · doi:10.1016/j.tcs.2007.03.031
[5] Beauquier, D., Nivat, M.: On translating one polyomino to tile the plane. Discrete Comput. Geom. 6, 575–592 (1991) · Zbl 0754.05030 · doi:10.1007/BF02574705
[6] Berthé, V., Fernique., T.: Brun expansions of stepped surfaces (Preprint) · Zbl 1236.11011
[7] Berthé, V., Vuillon, L.: Tilings and Rotations on the Torus: A Two-Dimensional Generalization of Sturmian Sequences. Discrete Math. 223, 27–53 (2000) · Zbl 0970.68124 · doi:10.1016/S0012-365X(00)00039-X
[8] Berthé, V., Fiorio, C., Jamet, D., Philippe, F.: On some applications of generalized functionality for arithmetic discrete planes. Image and Vision Computing 25, 1671–1684 (2007) · doi:10.1016/j.imavis.2006.06.023
[9] Berthé, V., Lacasse, A., Paquin, G., Provençal, X.: Boundary words for arithmetic discrete planes generated by Jacobi-Perron algorithm (Preprint)
[10] Blondin Massé, A., Brlek, S., Garon, A., Labbé, S.: Christoffel and Fibonacci Tiles. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 68–79. Springer, Heidelberg (2009) · Zbl 1244.05053
[11] Brlek, S., Fédou, J.M., Provençal, X.: On the tiling by translation problem. Discrete Applied Mathematics 157, 464–475 (2009) · Zbl 1156.05014 · doi:10.1016/j.dam.2008.05.026
[12] Brlek, S., Koskas, M., Provençal, X.: A linear time and space algorithm for detecting path intersection. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 398–409. Springer, Heidelberg (2009) · Zbl 1176.68004 · doi:10.1007/978-3-642-04397-0
[13] Brlek, S., Lachaud, J.O., Provençal, X., Reutenauer, C.: Lyndon+Christoffel=Digitally Convex. Pattern Recognition 42, 2239–2246 (2009) · Zbl 1176.68175 · doi:10.1016/j.patcog.2008.11.010
[14] Brentjes, A.J.: Mathematical Centre Tracts. Multi-dimensional continued fraction algorithms 145, Matematisch Centrum, Amsterdam (1981) · Zbl 0471.10024
[15] Brimkov, V., Coeurjolly, D., Klette, R.: Digital Planarity - A Review. Discr. Appl. Math. 155, 468–495 (2007) · Zbl 1109.68122 · doi:10.1016/j.dam.2006.08.004
[16] Baake, M., Moody, R.V., Robert, V. (eds.): Directions in mathematical quasicrystals. CRM Monograph Series, vol. 13. American Mathematical Society, Providence (2000) · Zbl 0955.00025
[17] Domenjoud, E., Jamet, D., Toutant, J.-L.: On the connecting thickness of arithmetical discrete planes. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 362–372. Springer, Heidelberg (2009) · Zbl 1261.68126 · doi:10.1007/978-3-642-04397-0_31
[18] Ei, H.: Some properties of invertible substitutions of rank d and higher dimensional substitutions. Osaka J. Math. 40, 543–562 (2003) · Zbl 1037.20033
[19] Fernique, T.: Multi-dimensional Sequences and Generalized Substitutions. Int. J. Fond. Comput. Sci. 17, 575–600 (2006) · Zbl 1096.68125 · doi:10.1142/S0129054106004005
[20] Fernique, T.: Generation and recognition of digital planes using multi-dimensional continued fractions. Pattern Recognition 432, 2229–2238 (2009) · Zbl 1176.68180 · doi:10.1016/j.patcog.2008.11.003
[21] Grunbaum, B., Shepard, G.: Tilings and patterns. Freeman, New-York (1987)
[22] Ito, S., Ohtsuki, M.: Modified Jacobi-Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms. Tokyo J. Math. 16, 441–472 (1993) · Zbl 0805.11056 · doi:10.3836/tjm/1270128497
[23] Jamet, D., Toutant, J.-L.: Minimal arithmetic thickness connecting discrete planes. Discrete Applied Mathematics 157, 500–509 (2009) · Zbl 1169.68050 · doi:10.1016/j.dam.2008.05.027
[24] Kenyon, R., Okounkov, O.: What is a dimer? Notices Amer. Math. Soc. 52, 342–343 (2005) · Zbl 1142.82339
[25] Lothaire, N.: Algebraic combinatorics on words. Cambridge University Press, Cambridge (2002) · Zbl 1001.68093 · doi:10.1017/CBO9781107326019
[26] Pytheas Fogg, N.: Substitutions in Dynamics, Arithmetics, and Combinatorics. In: Berthé, V., Ferenczi, S., Mauduit, C., Siegel, A. (eds.) Frontiers of Combining System. Lecture Notes in Mathematics, vol. 1794, Springer, Heidelberg (2002) · Zbl 1014.11015 · doi:10.1007/b13861
[27] Reveillès, J.-P.: Calcul en Nombres Entiers et Algorithmique. Thèse d’état, Université Louis Pasteur, Strasbourg, France (1991)
[28] Schweiger, F.: Multi-dimensional continued fractions. Oxford Science Publications, Oxford Univ. Press, Oxford (2000) · Zbl 0981.11029
[29] Senechal, M.: Quasicrystals and geometry. Cambridge University Press, Cambridge (1995) · Zbl 0828.52007
[30] Slater, N.B.: Gaps and steps for the sequence n \(\theta\) mod 1. Proc. Cambridge Phil. Soc. 63, 1115–1123 (1967) · Zbl 0178.04703 · doi:10.1017/S0305004100042195
[31] Thurston, W.P.: Groups, tilings and finite state automata. In: AMS Colloquium lectures. Lectures notes distributed in conjunction with the Colloquium Series (1989)
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