Geodesic
Metallic mean Wang tiles I: self-similarity, aperiodicity and minimality
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e133

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For every positive integer n, we introduce a set ${\mathcal {T}}_n$ made of $(n+3)^2$ Wang tiles (unit squares with labeled edges). We represent a tiling by translates of these tiles as a configuration $\mathbb {Z}^2\to {\mathcal {T}}_n$. A configuration is valid if the common edge of adjacent tiles has the same label. For every $n\geq 1$, we show that the Wang shift ${\Omega }_n$, defined as the set of valid configurations over the tiles ${\mathcal {T}}_n$, is self-similar, aperiodic and minimal for the shift action. We say that $\{{\Omega }_n\}_{n\geq 1}$ is a family of metallic mean Wang shifts, since the inflation factor of the self-similarity of $\Omega _n$ is the positive root of the polynomial $x^2-nx-1$. This root is sometimes called the n-th metallic mean, and in particular, the golden mean when $n=1$, and the silver mean when $n=2$. When $n=1$, the set of Wang tiles ${\mathcal {T}}_1$ is equivalent to the Ammann aperiodic set of 16 Wang tiles. 
Labbé, Sébastien. Metallic mean Wang tiles I: self-similarity, aperiodicity and minimality. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e133. doi: 10.1017/fms.2025.10069
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[1] Akiyama, S., ‘A note on aperiodic Ammann tiles’, Discrete Comput. Geom. 48(3) (2012), 702–710.10.1007/s00454-012-9418-4 Google Scholar | DOI

[2] Akiyama, S. and Araki, Y., ‘An alternative proof for an aperiodic monotile’, Preprint, 2023, . Google Scholar

[3] Akiyama, S. and Komornik, V., ‘Discrete spectra and Pisot numbers’, J. Number Theory 133(2) (2013), 375–390.10.1016/j.jnt.2012.07.015 Google Scholar | DOI

[4] Ammann, R., Grünbaum, B. and Shephard, G. C., ‘Aperiodic tiles’, Discrete Comput. Geom. 8(1) (1992), 1–25.10.1007/BF02293033 Google Scholar | DOI

[5] Baake, M. and Grimm, U.. Aperiodic Order. Vol. 1 (Encyclopedia of Mathematics and Its Applications) vol. 149 (Cambridge University Press, Cambridge, 2013). Google Scholar

[6] Baake, M., Gähler, F. and Sadun, L., ‘Dynamics and topology of the Hat family of tilings’, Preprint, 2023, . Google Scholar

[7] Beenker, F. P. M., Algebraic Theory of Non-Periodic Tilings of the Plane by Two Simple Building Blocks: A Square and a Rhombus (EUT-Rep.) (Eindhoven 82-WSK-04, 1982). Google Scholar

[8] Berger, R., ‘The undecidability of the domino problem’, Mem. Amer. Math. Soc. No. 66 (1966), 72. Google Scholar

[9] Berthé, V., Steiner, W., Thuswaldner, J. M. and Yassawi, R., ‘Recognizability for sequences of morphisms’, Ergodic Theory Dynam. Systems 39(11) (2019), 2896–2931.10.1017/etds.2017.144 Google Scholar | DOI

[10] Borwein, P. and Hare, K. G., ‘General forms for minimal spectral values for a class of quadratic Pisot numbers’, Bull. London Math. Soc. 35(1) (2003). 47–54.10.1112/S0024609302001455 Google Scholar | DOI

[11] Charlier, E., Kärki, T. and Rigo, M., ‘Multidimensional generalized automatic sequences and shape-symmetric morphic words’, Discrete Math. 310(6–7) (2010), 1238–1252.10.1016/j.disc.2009.12.002 Google Scholar | DOI

[12] Govert De Bruijn, N., ‘Algebraic theory of Penrose’s nonperiodic tilings of the plane, I, II’, Nederl. Akad. Wetensch. Indag. Math. 43(1) (1981), 39–52, 53–66.10.1016/1385-7258(81)90016-0 Google Scholar | DOI

[13] De Spinadel, V. W., ‘The family of metallic means,’ Vis. Math. 1(3) (1999), 1 HTML document; approx. 16. Google Scholar

[14] Dotera, T., Bekku, S. and Ziherl, P., ‘Bronze-mean hexagonal quasicrystal’, Nature Materials 16(10) (2017), 987–992.10.1038/nmat4963 Google Scholar PubMed | DOI

[15] Durand, B., Shen, A. and Vereshchagin, N., ‘On the structure of Ammann A2 tilings’, Discrete Comput. Geom. 63(3) (2020), 577–606.10.1007/s00454-019-00074-1 Google Scholar | DOI

[16] Durand, F., ‘A characterization of substitutive sequences using return words’, Discrete Math. 179(1–3) (1998), 89–101.10.1016/S0012-365X(97)00029-0 Google Scholar | DOI

[17] Frettlöh, D., Say-Awen, A. L. D. and De Las Peñas, M. L. A. N., ‘Substitution tilings with dense tile orientations and -fold rotational symmetry’, Indag. Math., New Ser. 28(1) (2017), 120–131.10.1016/j.indag.2016.11.009 Google Scholar | DOI

[18] Frettlöh, D., ‘More inflation tilings’, In Aperiodic order Vol. 2. (Encyclopedia Math. Appl.) vol. 166 (Cambridge Univ. Press, Cambridge, 2017), 1–37. Google Scholar

[19] Frettlöh, D., Garber, A. and Mañibo, N., ‘Substitution tilings with transcendental inflation factor’, Discrete Anal. (2024), Paper No. 11, 24. Google Scholar

[20] Frettlöh, D., Harriss, E. and Gähler, F., ‘Tilings encyclopedia: Bronze-mean tiling’, 2023, https://tilings.math.uni-bielefeld.de/substitution/bronze-mean/.10.1515/dmvm-2023-0006 Google Scholar | DOI

[21] Gähler, F., Julien, A. and Savinien, J, ‘Combinatorics and topology of the Robinson tiling’, C. R. Math. Acad. Sci. Paris 350(11–12) (2012), 627–631.10.1016/j.crma.2012.06.007 Google Scholar | DOI

[22] Gähler, F., Kwan, E. E. and Maloney, G. R., ‘A computer search for planar substitution tilings with -fold rotational symmetry’, Discrete Comput. Geom. 53(2) (2015), 445–465. Google Scholar | DOI

[23] Goodman-Strauss, C., ‘Matching rules and substitution tilings’, Ann. of Math. (2) 147(1) (1998), 181–223.10.2307/120988 Google Scholar | DOI

[24] Grünbaum, B. and Shephard, G. C., Tilings and Patterns (W. H. Freeman and Company, New York, 1987). Google Scholar

[25] Hochman, M., ‘Multidimensional shifts of finite type and sofic shifts’, in Combinatorics, Words and Symbolic Dynamics (Encyclopedia Math. Appl.) vol. 159 (Cambridge Univ. Press, Cambridge, 2016), 296–358. Google Scholar

[26] Jeandel, E. and Rao, M., ‘An aperiodic set of 11 Wang tiles’, Adv. Comb. 2021 (2021), 37. Id/No 1. Google Scholar

[27] Kari, J. and Papasoglu, P., ‘Deterministic aperiodic tile sets’, Geom. Funct. Anal. 9(2) (1999), 353–369.10.1007/s000390050090 Google Scholar | DOI

[28] Kari, J. and Lutfalla, V. H., ‘Substitution discrete plane tilings with -fold rotational symmetry for odd ’, Discrete Comput. Geom. 69(2) (2023), 349–398.10.1007/s00454-022-00390-z Google Scholar | DOI

[29] Kari, J. and Rissanen, M., ‘Sub Rosa, a system of quasiperiodic rhombic substitution tilings with -fold rotational symmetry’, Discrete Comput. Geom. 55(4) (2016), 972–996. Google Scholar | DOI

[30] Knuth, D. E., The Art of Computer Programming. Vol. 1: Fundamental Algorithms, second printing (Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont, 1969). Google Scholar

[31] Knuth, D. E., ‘Dancing links’, in Millenial Perspectives in Computer Science (Red Globe Press, London, 2000), 187–214, . Google Scholar

[32] Komatsu, T., ‘An approximation property of quadratic irrationals’, Bull. Soc. Math. France 130(1) (2002), 35–48. Google Scholar | DOI

[33] Labbé, S., ‘A self-similar aperiodic set of 19 Wang tiles’, Geom. Dedicata 201 (2019), 81–109. Google Scholar | DOI

[34] Labbé, S., ‘Markov partitions for toral -rotations featuring Jeandel-Rao Wang shift and model sets’, Ann. H. Lebesgue 4 (2021), 283–324.10.5802/ahl.73 Google Scholar | DOI

[35] Labbé, S., ‘Substitutive structure of Jeandel-Rao aperiodic tilings’, Discrete Comput. Geom. 65(3) (2021), 800–855. Google Scholar | DOI

[36] Labbé, S., Mann, C. and Mcloud-Mann, J., ‘Nonexpansive directions in the Jeandel-Rao Wang shift’, Discrete Contin. Dyn. Syst. 43(9) (2023), 3213–3250. Google Scholar | DOI

[37] Labbé, S., ‘Three characterizations of a self-similar aperiodic 2-dimensional subshift’, Preprint, 2020, . Google Scholar

[38] Labbé, S., Optional SageMath Package slabbe (Version 0.7.7), 2024, https://pypi.python.org/pypi/slabbe/. Google Scholar

[39] Lind, D., ‘Multi-dimensional symbolic dynamics’, in Symbolic Dynamics and Its Applications (Proc. Sympos. Appl. Math.) vol. 60 (Amer. Math. Soc., Providence, RI, 2004), 61–79.10.1090/psapm/060/2078846 Google Scholar | DOI

[40] Lind, D. and Marcus, B., An Introduction to Symbolic Dynamics and Coding (Cambridge University Press, Cambridge, 1995). Google Scholar | DOI

[41] Masáková, Z., Pastirčáková, K. and Pelantová, E., ‘Description of spectra of quadratic Pisot units’, J. Number Theory 150 (2015), 168–190.10.1016/j.jnt.2014.11.011 Google Scholar | DOI

[42] Mossé, B., ‘Puissances de mots et reconnaissabilité des points fixes d’une substitution’, Theoret. Comput. Sci. 99(2) (1992), 327–334. Google Scholar | DOI

[43] Mozes, S., ‘Tilings, substitution systems and dynamical systems generated by them’, J. Analyse Math. 53 (1989), 139–186. Google Scholar | DOI

[44] Nakakura, J., Ziherl, P., Matsuzawa, J. and Dotera, T., ‘Metallic-mean quasicrystals as aperiodic approximants of periodic crystals’, Nature Communications 10(1) (2019), Article number: 4235.10.1038/s41467-019-12147-z Google Scholar PubMed | DOI

[45] OEIS Foundation Inc., ‘Entry A352403 in the on-line encyclopedia of integer sequences’, 2023, https://oeis.org/A352403. Google Scholar

[46] OEIS Foundation Inc., ‘Metallic means’, 2023, https://oeis.org/wiki/Metallic_means. Google Scholar

[47] Pautze, S., ‘Cyclotomic aperiodic substitution tilings’, Symmetry 9(2) (2017), Article number: 19. Google Scholar | DOI

[48] Penrose, R., ‘The rôle of aesthetics in pure and applied mathematical research’, Bull. Inst. Math. Appl. 10(Jul–Aug) (1974), 266–271. Google Scholar

[49] Penrose, R., ‘Pentaplexity. A class of non-periodic tilings of the plane’, Math. Intell. 2 (1979), 32–37.10.1007/BF03024384 Google Scholar | DOI

[50] Penrose, R., ‘Remarks on tiling: Details of a -aperiodic set’, in The Mathematics of Long-Range Aperiodic Order. Proceedings of the NATO Advanced Study Institute, Waterloo, Ontario, Canada, August 21–September 1, 1995 (Kluwer Academic Publishers, Dordrecht, 1997), 467–497. Google Scholar

[51] Queffélec, M., Substitution Dynamical Systems—Spectral Analysis (Lecture Notes in Mathematics) vol. 1294, second edn. (Springer-Verlag, Berlin, 2010).10.1007/978-3-642-11212-6 Google Scholar | DOI

[52] Robinson, E. A., ‘On the table and the chair’, Indag. Math., New Ser. 10(4) (1999), 581–599.10.1016/S0019-3577(00)87911-2 Google Scholar | DOI

[53] Robinson, R. M., ‘Undecidability and nonperiodicity for tilings of the plane’, Invent. Math. 12 (1971), 177–209.10.1007/BF01418780 Google Scholar | DOI

[54] Developers, Sage, SageMath, the Sage Mathematics Software System (Version 10.5), 2024, http://www.sagemath.org. Google Scholar

[55] Schmidt, K., ‘Multi–dimensional symbolic dynamical systems’, in Codes, Systems, and Graphical Models (Minneapolis, MN, 1999) (IMA Vol. Math. Appl.) vol. 123 (Springer, New York, 2001), 67–82.10.1007/978-1-4613-0165-3_3 Google Scholar | DOI

[56] Schroeder, M., Fractals, Chaos, Laws, Power. Minutes from an Infinite Paradise (W.H. Freeman and Company, New York, 1991). Google Scholar

[57] Senechal, M., ‘The mysterious Mr. Ammann’, Math. Intelligencer 26(4) (2004), 10–21. Google Scholar | DOI

[58] Smith, D., Myers, J. S., Kaplan, C. S. and Goodman-Strauss, C., ‘An aperiodic monotile’, Comb. Theory 4(1) (2024), Paper No. 6, 91. Google Scholar

[59] Smith, D., Myers, J. S., Kaplan, C. S. and Goodman-Strauss, C., ‘A chiral aperiodic monotile’, Comb. Theory 4(2) (2024), Paper No. 13, 25. Google Scholar

[60] Socolar, J. E. S., ‘Quasicrystalline structure of the hat monotile tilings’, Phys. Rev. B 108 (2023), 224109.10.1103/PhysRevB.108.224109 Google Scholar | DOI

[61] Socolar, J. E. S. and Taylor, J. M., ‘An aperiodic hexagonal tile’, J. Combin. Theory Ser. A 118(8) (2011), 2207–2231. Google Scholar | DOI

[62] Solomyak, B., ‘Nonperiodicity implies unique composition for self-similar translationally finite tilings’, Discrete Comput. Geom. 20(2) (1998), 265–279.10.1007/PL00009386 Google Scholar | DOI

[63] Solomyak, B., ‘Dynamics of self-similar tilings’, Ergodic Theory Dynam. Systems 17(3) (1997), 695–738.10.1017/S0143385797084988 Google Scholar | DOI

[64] Vuillon, L., ‘A characterization of Sturmian words by return words’, European J. Combin. 22(2) (2001), 263–275. Google Scholar | DOI

[65] Walters, P., An Introduction to Ergodic Theory (GTM) vol. 79 (Springer-Verlag, New York-Berlin, 1982).10.1007/978-1-4612-5775-2 Google Scholar | DOI

[66] Wang, H., ‘Proving theorems by pattern recognition – II’, Bell System Technical Journal 40(1) (1961), 1–41.10.1002/j.1538-7305.1961.tb03975.x Google Scholar | DOI

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