Abstract
A substitution tiling is a certain globally defined hierarchical structure in a geometric space. In [6] we show that for any substitution tiling in En, n > 1, subject to relatively mild conditions, one can construct local rules that force the desired global structure to emerge. As an immediate corollary, infinite collections of forced aperiodic tilings are constructed. Here we give an expository account of the construction. In particular, we discuss the use of hierarchical, algorithmic, geometrically sensitive coordinates—“addresses”, developed further in [9].
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References
R. Berger, The undecidability of the domino problem, Memoirs Am. Math. Soc. 66 (1966).
N.C. de Bruijn, Algebraic theory of Penrose’s non-periodic tilings,Nederl. Akad. Wentensch Proc. Ser. A 84 (1981), 39–66.
L. Danzer, A family of SD-spacefillers not permitting any periodic or quasiperiodic tilingpreprint
L. Danzer, personal communication.
C. Godreche, The sphinx: a limit periodic tiling of the plane, J. F’hys. A: Math. Gen. 22 1989, L1163 - L1166.
C. Goodman-Strauss, Matching rules and substitution tilings,to appear in Aimais of Math.
C. Goodman-Strauss, A small aperiodic set of tiles, preprint.
C. Goodman-Strauss, An aperiodic tiling of E“ for all n 1, preprint.
C Goodman-Strauss, Addresses and substitution tidings, in preparation.
B. Grunbaum and G.C. Shepherd, Tilings and patterns, W.H. Freeman and Co. (1989).
J. Kari, A small aperiodic set of Wang tiles, preprint.
T.T.Q. Le, Local rules for quasiperiodic tidings, Proceedings of the Fields Institute (1995).
J.M. Luck, Aperiodic structures: geometry, diffraction spectra, and physical properties, to appear in Fund. Prob. in Stat. Mech VIII.
S. Mozes, Tilings, substitution systems and dynamical systems generated by them, J. D’Analyse Math. 53 (1989), 139–186.
R. Penrose, The role of aesthetics in pure and applied mathematical research, Bull. Inst. of Math. and its Appl. 10 (1974) 266–71.
P. Prusinkiewicz and A. Lindenmeyer, The Algorithmic Beauty of Plants, Springer-Verlag (1990).
Radin, The pinwheel things of the plane, Annals of Math. 139 (1994), 661–702.
C. Radin, personal communication.
E.A. Robinson, personal communication.
R. Robinson, Undecidability and nonperiodicity of tidings in the plane, Inv. Math. 12 (1971), 177.
L. Sadun, Some generalizations of the pinwheel tiling, preprint.
M. Senechal, Quasicrystals and geometry, Cambridge University Press (1995).
J.E.S Socolar, personal communication.
J.E.S. Socolar and P.J. Steinhardt, Quasicrystals II. Unit cell configurations, Physical Review B 34 (1986), 617–647.
B. Solomyak, Non-periodicity implies unique composition for self-similar translationally-finite tilings, to appear in J.Disc. and Comp. Geometry.
P.J. Steinhardt and S. Ostlund, The physics of quasicrystals, World Scientific (1987).
W. Thurston, Croups, tilings and finite state automata: Summer 1989 AMS collo-quint lectures, GCG 1, Geometry Center.
P. Walters, editor, Symbolic dynamics and its applications, Contemporary Mathematics 135 (1992).
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© 1999 Springer Science+Business Media Dordrecht
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Goodman-Strauss, C. (1999). Aperiodic Hierarchical Tilings. In: Sadoc, J.F., Rivier, N. (eds) Foams and Emulsions. NATO ASI Series, vol 354. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9157-7_28
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DOI: https://doi.org/10.1007/978-94-015-9157-7_28
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