Tiling Spaces of Taylor-Socolar Tilings

• Authors: J. Lee
• Languages of publication: EN

Abstracts

EN

The Taylor-Socolar tiling has been introduced as an aperiodic mono-tile tiling. We consider a tiling space which consists of all the tilings that are locally indistinguishable from a Taylor-Socolar tiling and study its structure. It turns out that there is a bijective map between the space of the Taylor-Socolar tilings and a compact Abelian group of a Q-adic space (Q̅) except at a dense set of points of measure 0 in Q̅. From this we can derive that the Taylor-Socolar tilings have quasicrystalline structures. We make a parity tiling from the Taylor-Socolar tiling identifying all the rotated versions of a tile in the Taylor-Socolar tiling by white tiles and all the reflected versions of the tile by gray tiles. It turns out that the Taylor-Socolar tiling is mutually locally derivable from this parity tiling.

Keywords

EN

61.44.Br; 61.44.-n

Discipline

• 61.44.-n: Semi-periodic solids
• 61.44.Br: Quasicrystals

• Publisher: Institute of Physics, Polish Academy of Sciences
• Journal: Acta Physica Polonica A
• Year: 2014
• Volume: 126
• Issue: 2
• Pages: 508-511

Dates

published

2014-08

Contributors

author

J. Lee

• Department of Mathematics Education, Kwandong University, Gangneung, Gyeonggi-do, 210-701, Korea

References

• [1] J. Socolar, J. Taylor, J. Combinat. Theor., Series A 118, 2207 (2011), doi: 10.1016/j.jcta.2011.05.001
• [2] J. Socolar, J. Taylor, Math. Intelligencer 34, 18 (2012), doi: 10.1007/s00283-011-9255-y
• [3] R. Penrose, Twistor Newsletter, reprinted in Roger Penrose: Collected Works, Vol. 6, 1997-2003, Oxford University Press, New York 2010 http://people.maths.ox.ac.uk/lmason/Tn/
• [4] R. Penrose, in: The Mathematics of Long-Range Aperiodic Order, Ed. R.V. Moody, NATO ASI Series C, Vol. 489, Kluwer Acad. Publ., Dordrecht (Netherlands) 1997, p. 467
• [5] M. Baake, F. Gähler, U. Grimm, Symmetry 4, 581 (2012), doi: 10.3390/sym4040581
• [6] F. Gähler, private communication
• [7] J.-Y. Lee, R.V. Moody, 'A Relation between Two Aperiodic Hexagonal Tilings', preprint
• [8] C. Radin, M. Wolff, Geometr. Dedic. 42, 355 (1992), doi: 10.1007/BF02414073
• [9] J.-Y. Lee, R.V. Moody, Symmetry 5, 1 (2013), doi: 10.3390/sym5010001

Identifiers

DOI

10.12693/APhysPolA.126.508