Abstract
On the one hand, Socolar showed in 1990 that the n-fold planar tilings admit weak local rules when n is not divisible by 4 (the \(n=10\) case corresponds to the Penrose tilings and is known since 1974). On the other hand, Burkov showed in 1988 that the eightfold tilings do not admit weak local rules, and Le showed the same for the 12-fold tilings (unpublished). We here show that this is actually the case for all the 4p-fold tilings.
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Ammann, R., Grünbaum, B., Shephard, G.C.: Aperiodic tiles. Discrete Comput. Geom. 8, 1–25 (1992)
Bédaride, N., Fernique, Th: Ammann–Beenker tilings revisited. In: Schmid, S., Withers, R.L., Lifshitz, R. (eds.) Aperiodic Crystals, pp. 59–65. Springer, Dordrecht (2013)
Bédaride, N., Fernique, Th: When periodicities enforce aperiodicity. Commun. Math. Phys. 335, 1099–1120 (2015)
Burkov, S.E.: Absence of weak local rules for the planar quasicrystalline tiling with the 8-fold rotational symmetry. Commun. Math. Phys. 119, 667–675 (1988)
Fernique, Th., Sablik, M.: Local rules for computable planar tilings (preprint)
Julien, A.: Complexity and cohomology for cut-and-projection tilings. Ergodic Theory Dyn. Syst. 30, 489–523 (2010)
Katz, A.: Matching rules and quasiperiodicity: the octagonal tilings. In: Axel, F., Gratias, D. (eds.) Beyond Quasicrystals, pp. 141–189. Springer, Berlin (1995)
Kleman, M., Pavlovitch, A.: Generalized 2D Penrose tilings: structural properties. J. Phys. A: Math. Gen. 20, 687–702 (1987)
Le, T.Q.T., Piunikhin, S.A., Sadov, V.A.: Local rules for quasiperiodic tilings of quadratic 2-planes in \({\mathbb{R}}^4\). Commun. Math. Phys. 150, 23–44 (1992)
Le, T.Q.T.: Local structure of quasiperiodic tilings having 8-fold symmetry (preprint) (1992)
Le, T.Q.T.: Necessary conditions for the existence of local rules for quasicrystals (preprint) (1992)
Le, T.Q.T., Piunikhin, S.A., Sadov, V.A.: The geometry of quasicrystals. Russ. Math. Surv. 48, 37–100 (1993)
Le, T.Q.T.: Local rules for pentagonal quasi-crystals. Discrete Comput. Geom. 14, 31–70 (1995)
Le, T.Q.T.: Local rules for quasiperiodic tilings in the mathematics long range aperiodic order. NATO Adv. Sci. Inst. Ser. C. Math. Phys. Sci. 489, 331–366 (1995)
Levitov, L.S.: Local rules for quasicrystals. Commun. Math. Phys. 119, 627–666 (1988)
Robinson, A.: Symbolic dynamics and tilings of \({\mathbb{R}}^d\). Symbolic dynamics and its applications. In: Proc. Sympos. Appl. Math., vol. 60, pp. 81–119. American Mathematical Society, Providence, RI (2004)
Senechal, M.: Quasicrystals and Geometry. Cambridge University Press, Cambridge (1995)
Socolar, J.E.S.: Simple octagonal and dodecagonal quasicrystals. Phys. Rev. B 39, 10519–10551 (1989)
Socolar, J.E.S.: Weak matching rules for quasicrystals. Commun. Math. Phys. 129, 599–619 (1990)
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This work was supported by the ANR project QuasiCool (ANR-12-JS02-011-01).
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Bédaride, N., Fernique, T. No Weak Local Rules for the 4p-Fold Tilings. Discrete Comput Geom 54, 980–992 (2015). https://doi.org/10.1007/s00454-015-9740-8
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DOI: https://doi.org/10.1007/s00454-015-9740-8