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Canonical projection tilings defined by patterns. (English) Zbl 1450.37017

Summary: We give a necessary and sufficient condition on a \(d\)-dimensional affine subspace of \(\mathbb{R}^n\) to be characterized by a finite set of patterns which are forbidden to appear in its digitization. This can also be stated in terms of local rules for canonical projection tilings, or subshift of finite type. This provides a link between algebraic properties of affine subspaces and combinatorics of their digitizations. The condition relies on the notion of coincidence and can be effectively checked. As a corollary, we get that only algebraic subspaces can be characterized by patterns.

MSC:

37B52 Tiling dynamics
37B51 Multidimensional shifts of finite type

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[1] Ammann, R.; Grünbaum, B.; Shephard, GC, Aperiodic tiles, Discrete Comput. Geom., 8, 1-25 (1992) · Zbl 0758.52013 · doi:10.1007/BF02293033
[2] Beenker, F.P.M.: Algebric theory of non periodic tilings of the plane by two simple building blocks: a square and a rhombus. Technical Report TH Report 82-WSK-04, Technische Hogeschool Eindhoven (1982) · Zbl 0545.05033
[3] Bédaride, N., Fernique, Th.: Aperiodic Crystals. In: The Ammann-Beenker Tilings Revisited, pp. 59-65. Springer, Dordrecht (2013)
[4] Bédaride, N.; Fernique, Th, No weak local rules for the 4p-fold tilings, Discrete Comput. Geom., 54, 980-992 (2015) · Zbl 1335.52029 · doi:10.1007/s00454-015-9740-8
[5] Bédaride, N.; Fernique, Th, When periodicities enforce aperiodicity, Commun. Math. Phys., 335, 1099-1120 (2015) · Zbl 1315.52017 · doi:10.1007/s00220-015-2334-8
[6] Bédaride, N.; Fernique, Th, Weak local rules for octagonal tilings, Israel J. Math., 222, 63-89 (2017) · Zbl 1382.52019 · doi:10.1007/s11856-017-1582-z
[7] Baake, M.; Grimm, U., Aperiodic Order. Encyclopedia of Mathematics and Its Applications (2013), Cambridge: Cambridge University Press, Cambridge · Zbl 1295.37001
[8] Burkov, SE, Absence of weak local rules for the planar quasicrystalline tiling with the \(8\)-fold rotational symmetry, Commun. Math. Phys., 119, 667-675 (1988) · Zbl 0655.05025 · doi:10.1007/BF01218349
[9] De Bruijn, NG, Algebraic theory of Penrose’s nonperiodic tilings of the plane, Nederl. Akad. Wetensch. Indag. Math., 43, 39-52 (1981) · Zbl 0457.05021 · doi:10.1016/1385-7258(81)90016-0
[10] Fernique, Th., Sablik, M.: Weak colored local rules for planar tilings. Ergod. Theor. Dyn. Syst. (2018) · Zbl 1439.52021
[11] Grünbaum, B.; Shephard, GC, Tilings and Patterns (1986), New York, NY: W. H. Freeman & Co., New York, NY
[12] Haynes, A., Julien, A., Koivusalo, H., Walton, J.: Statistics of patterns in typical cut and project sets. Ergod. Theor. Dyn. Syst., 1-23 (2018) · Zbl 1448.37010
[13] Haynes, A.; Koivusalo, H.; Sadun, L.; Walton, J., Gaps problems and frequencies of patches in cut and project sets, Math. Proc. Camb. Philos. Soc., 161, 65-85 (2016) · Zbl 1371.11110 · doi:10.1017/S0305004116000128
[14] Hodge, WVD; Pedoe, D., Methods of Algebraic Geometry (1994), Cambridge: Cambridge University Press, Cambridge · Zbl 0796.14001
[15] Julien, A., Complexity and cohomology for cut-and-projection tilings, Ergod. Theor. Dyn. Syst., 30, 489-523 (2010) · Zbl 1185.37031 · doi:10.1017/S0143385709000194
[16] Katz, A., Theory of matching rules for the 3-dimensional Penrose tilings, Commun. Math. Phys., 118, 263-288 (1988) · Zbl 0651.52015 · doi:10.1007/BF01218580
[17] Katz, A.: Beyond Quasicrystals: Les Houches, March 7-18, 1994, Chapter Matching Rules and Quasiperiodicity: The Octagonal Tilings, pp. 141-189. Springer, Berlin (1995) · Zbl 0882.52011
[18] Kleman, M.; Pavlovitch, A., Generalised 2d Penrose tilings: structural properties, J. Phys. A: Math. Gen., 20, 687-702 (1987) · Zbl 0653.05018 · doi:10.1088/0305-4470/20/3/031
[19] Le, TQT, Local rules for pentagonal quasi-crystals, Discrete Comput. Geom., 14, 31-70 (1995) · Zbl 0842.52011 · doi:10.1007/BF02570695
[20] Le, T.Q.T.: Local rules for quasiperiodic tilings. In: The Mathematics of Long-Range Aperiodic Order (Waterloo, ON, 1997), Volume 489 of NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, pp. 331-366. Kluwer Academy Publication, Dordrecht (1997) · Zbl 0884.52019
[21] Levitov, LS, Local rules for quasicrystals, Commun. Math. Phys., 119, 627-666 (1988) · doi:10.1007/BF01218348
[22] Le, TQT; Piunikhin, S., Local rules for multi-dimensional quasicrystals, Differ. Geom. Appl., 5, 10-31 (1995) · Zbl 0816.52006
[23] Le, TQT; Piunikhin, S.; Sadov, V., Local rules for quasiperiodic tilings of quadratic \(2\)-planes in \({ R}^4\), Commun. Math. Phys., 150, 23-44 (1992) · Zbl 0769.52016 · doi:10.1007/BF02096563
[24] Penrose, R.: Pentaplexity: a class of non-periodic tilings of the plane. Eureka 39, 16-22 (1978)
[25] Senechal, M., Quasicrystals and Geometry (1995), Cambridge: Cambridge University Press, Cambridge · Zbl 0828.52007
[26] Socolar, JES, Weak matching rules for quasicrystals, Commun. Math. Phys., 129, 599-619 (1990) · Zbl 0701.05059 · doi:10.1007/BF02097107
[27] The Sage Developers: SageMath, the Sage Mathematics Software System (Version 7.1) (2016). http://www.sagemath.org Accessed 2 Feb 2020
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