Osaka Journal of Mathematics
- Osaka J. Math.
- Volume 51, Number 3 (2014), 597-609.
The computation of overlap coincidence in Taylor--Socolar substitution tiling
Shigeki Akiyama and Jeong-Yup Lee
Full-text: Open access
Abstract
Recently Taylor and Socolar introduced an aperiodic mono-tile. The associated tiling can be viewed as a substitution tiling. We use the substitution rule for this tiling and apply the algorithm of [1] to check overlap coincidence. It turns out that the tiling has overlap coincidence. So the tiling dynamics has pure point spectrum and we can conclude that this tiling has a quasicrystalline structure.
Article information
Source
Osaka J. Math., Volume 51, Number 3 (2014), 597-609.
Dates
First available in Project Euclid: 23 October 2014
Permanent link to this document
https://projecteuclid.org/euclid.ojm/1414090793
Mathematical Reviews number (MathSciNet)
MR3272607
Zentralblatt MATH identifier
06368439
Subjects
Primary: 52C23: Quasicrystals, aperiodic tilings
Citation
Akiyama, Shigeki; Lee, Jeong-Yup. The computation of overlap coincidence in Taylor--Socolar substitution tiling. Osaka J. Math. 51 (2014), no. 3, 597--609. https://projecteuclid.org/euclid.ojm/1414090793
References
- S. Akiyama and J.-Y. Lee: Algorithm for determining pure pointedness of self-affine tilings, Adv. Math. 226 (2011), 2855–2883. Mathematical Reviews (MathSciNet): MR2764877
Digital Object Identifier: doi:10.1016/j.aim.2010.07.019 - R. Ammann, B. Grünbaum and G.C. Shephard: Aperiodic tiles, Discrete Comput. Geom. 8 (1992), 1–25. Mathematical Reviews (MathSciNet): MR1156132
- S. Akiyama and J.-Y. Lee: Algorithm to compute overlap coincidence of Taylor–Socolar tiling, http://math.tsukuba.ac.jp/~ akiyama/Research1.html, http:// newton.kias.re.kr/~ jeongyup/Research/
- \begingroup R. Berger: The undecidability of the domino problem, Mem. Amer. Math. Soc. No. 66 (1966), 72. \endgroup Mathematical Reviews (MathSciNet): MR216954
- M. Baake and F. Gähler: Oral communications.
- M. Baake, F. Gähler and U. Grimm: Hexagonal inflation tilings and planar monotiles, Symmetry 4 (2012), 581–602.
- R.M. Robinson: Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12 (1971), 177–209.
- M. Baake and D. Lenz: Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra, Ergodic Theory Dynam. Systems 24 (2004), 1867–1893. Mathematical Reviews (MathSciNet): MR2106769
Digital Object Identifier: doi:10.1017/S0143385704000318 - M. Baake and R.V. Moody: Weighted Dirac combs with pure point diffraction, J. Reine Angew. Math. 573 (2004), 61–94. Mathematical Reviews (MathSciNet): MR2084582
- D. Frettlöh: Nichtperiodische Pflasterungen mit ganzzahligem Inflationsfaktor, Ph.D. Thesis, Univ. Dortmund, 2002.
- F. Gähler and R. Klitzing: The diffraction pattern of self-similar tilings; in The Mathematics of Long-Range Aperiodic Order (Waterloo, ON, 1995), Kluwer Acad. Publ., Dordrecht, 1977, 141–174. Mathematical Reviews (MathSciNet): MR1460023
- M. Gardner: Extraordinary nonperiodic tiling that enriches the theory of tiles, Sci. Amer. (USA) 236 (1977), 110–119
- J.-B. Gouéré: Diffraction et mesure de Palm des processus ponctuels, C.R. Math. Acad. Sci. Paris 336 (2003), 57–62. Mathematical Reviews (MathSciNet): MR1968903
Digital Object Identifier: doi:10.1016/S1631-073X(02)00029-8 - B. Grünbaum and G.C. Shephard: Tilings and Patterns, Freeman, New York, 1987. Mathematical Reviews (MathSciNet): MR857454
- A. Hof: Diffraction by aperiodic structures; in The Mathematics of Long-Range Aperiodic Order (Waterloo, ON, 1995), Kluwer Acad. Publ., Dordrecht, 1977, 239–268. Mathematical Reviews (MathSciNet): MR1460026
Digital Object Identifier: doi:10.1007/978-94-015-8784-6_10 - International Union of Crystallography: Report of the executive commitee for 1991, Acta Cryst. A48 (1992), 922–946.
- J.C. Lagarias: Meyer's concept of quasicrystal and quasiregular sets, Comm. Math. Phys. 179 (1996), 365–376. Mathematical Reviews (MathSciNet): MR1400744
Digital Object Identifier: doi:10.1007/BF02102593
Project Euclid: euclid.cmp/1104286996 - J.C. Lagarias and P.A.B. Pleasants: Repetitive Delone sets and quasicrystals, Ergodic Theory Dynam. Systems 23 (2003), 831–867. Mathematical Reviews (MathSciNet): MR1992666
Digital Object Identifier: doi:10.1017/S0143385702001566 - J.-Y. Lee: Substitution Delone sets with pure point spectrum are inter-model sets, J. Geom. Phys. 57 (2007), 2263–2285. Mathematical Reviews (MathSciNet): MR2360242
Digital Object Identifier: doi:10.1016/j.geomphys.2007.07.003 - J.-Y. Lee and R.V. Moody: Lattice substitution systems and model sets, Discrete Comput. Geom. 25 (2001), 173–201. Mathematical Reviews (MathSciNet): MR1811757
- J.-Y. Lee and R.V. Moody: Taylor–Socolar hexagonal tilings as model sets, Symmetry 5 (2013), 1–46.
- J.-Y. Lee, R.V. Moody and B. Solomyak: Pure point dynamical and diffraction spectra, Ann. Henri Poincaré 3 (2002), 1003–1018. Mathematical Reviews (MathSciNet): MR1937612
Digital Object Identifier: doi:10.1007/s00023-002-8646-1 - J.-Y. Lee, R.V. Moody and B. Solomyak: Consequences of pure point diffraction spectra for multiset substitution systems, Discrete Comput. Geom. 29 (2003), 525–560. Mathematical Reviews (MathSciNet): MR1976605
- D. Lenz and N. Strungaru: Pure point spectrum for measure dynamical systems on locally compact abelian groups, J. Math. Pures Appl. (9) 92 (2009), 323–341. Mathematical Reviews (MathSciNet): MR2569181
Digital Object Identifier: doi:10.1016/j.matpur.2009.05.013 - J. Morita: Tilings, Lie theory and combinatorics; in Quantum Affine Algebras, Extended Affine Lie Algebras, and Their Applications, Contemp. Math. 506, Amer. Math. Soc., Providence, RI, 2010, 173–185.
- J. Morita and A. Terui: Words, tilings and combinatorial spectra, Hiroshima Math. J. 39 (2009), 37–60.
- R. Penrose: Remarks on tiling: details of a $(1+\epsilon+\epsilon\sp 2)$-aperiodic set; in The Mathematics of Long-Range Aperiodic Order (Waterloo, ON, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 489, Kluwer Acad. Publ., Dordrecht, 1997, 467–497. Mathematical Reviews (MathSciNet): MR1460034
Digital Object Identifier: doi:10.1007/978-94-015-8784-6_18 - R. Penrose: Solutions to puzzles in TN41, Twistor Newsletter 42 (1997), 121–126.
- \begingroup C. Radin: Aperiodic tilings in higher dimensions, Proc. Amer. Math. Soc. 123 (1995), 3543–3548. \endgroup Mathematical Reviews (MathSciNet): MR1277129
Digital Object Identifier: doi:10.1090/S0002-9939-1995-1277129-X - M. Senechal: Quasicrystals and Geometry, Cambridge Univ. Press, Cambridge, 1995. Mathematical Reviews (MathSciNet): MR1340198
- D. Shechtman, I. Blech, D. Gratias and J.W. Cahn: Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53 (1984), 1951–1953. \interlinepenalty10000
- J.E.S. Socolar and J.M. Taylor: An aperiodic hexagonal tile, J. Combin. Theory Ser. A 118 (2011), 2207–2231. Mathematical Reviews (MathSciNet): MR2834173
Digital Object Identifier: doi:10.1016/j.jcta.2011.05.001 - J.E.S. Socolar and J.M. Taylor: Forcing nonperiodicity with a single tile, Math. Intelligencer 34 (2012), 18–28. Mathematical Reviews (MathSciNet): MR2902144
Digital Object Identifier: doi:10.1007/s00283-011-9255-y - B. Solomyak: Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems 17 (1997), 695–738. Mathematical Reviews (MathSciNet): MR1452190
Digital Object Identifier: doi:10.1017/S0143385797084988 - J. Taylor: Aperiodicity of a functional monotile, Preprint (2010), http:// www.math.uni-bielefeld.de/sfb701/files/preprints/sfb10015.pdf

- You have access to this content.
- You have partial access to this content.
- You do not have access to this content.
More like this
- On Disk-like Self-affine Tiles Arising from Polyominoes
Gmainer, Johannes and Thuswaldner, Jörg M., Methods and Applications of Analysis, 2006 - On the Shuffling Algorithm for Domino Tilings
Nordenstam, Eric, Electronic Journal of Probability, 2010 - A substitution rule for the Penrose tiling
Komatsu, Kazushi and Nakano, Fumihiko, Nihonkai Mathematical Journal, 2008
- On Disk-like Self-affine Tiles Arising from Polyominoes
Gmainer, Johannes and Thuswaldner, Jörg M., Methods and Applications of Analysis, 2006 - On the Shuffling Algorithm for Domino Tilings
Nordenstam, Eric, Electronic Journal of Probability, 2010 - A substitution rule for the Penrose tiling
Komatsu, Kazushi and Nakano, Fumihiko, Nihonkai Mathematical Journal, 2008 - Eigenfunctions for substitution tiling systems
Solomyak, Boris, , 2007 - A Modified Multifractal Formalism for a Class of Self-similar
Measures with Overlap
Shmerkin, Pablo, Asian Journal of Mathematics, 2005 - Vertices of self-similar tiles
Deng, Da-Wen and Ngai, Sze-Man, Illinois Journal of Mathematics, 2005 - The $K$-Group of Substitutional Systems
El Kacimi, Aziz and Parthasarathy, Rajagopalan, Publicacions Matemàtiques, 2010 - Decay rates for some quasi-birth-and-death processes with phase-dependent transition rates
Motyer, Allan J. and Taylor, Peter G., Journal of Applied Probability, 2011 - Completeness of MLL proof-nets w.r.t. weak distributivity
Joinet, Jean-Baptiste, Journal of Symbolic Logic, 2007 - Rule-Irredundancy and the Sequent Calculus for Core Logic
Tennant, Neil, Notre Dame Journal of Formal Logic, 2016