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Discrete Dynamics in Nature and Society
Volume 2011, Article ID 946913, 23 pages
http://dx.doi.org/10.1155/2011/946913
Research Article

Randomness and Topological Invariants in Pentagonal Tiling Spaces

Facultad de Ciencias, Universidad de Oviedo, 33007 Oviedo, Spain

Received 27 February 2011; Revised 25 April 2011; Accepted 26 April 2011

Academic Editor: Binggen Zhang

Copyright © 2011 Juan García Escudero. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. H. Wang, “Proving theorems by pattern recognition-II,” Bell System Technical Journal, vol. 40, pp. 1–41, 1961. View at Google Scholar
  2. R. Penrose, “Role of aesthetics in pure and applied research,” Bulletin of the Institute of Mathematics and Its Applications, vol. 10, article 266ff, 1974. View at Google Scholar
  3. C. Radin, Miles of Tiles, vol. 1 of Student Mathematical Library, American Mathematical Society, Providence, RI, USA, 1999.
  4. M. Baake, M. Birkner, and R. V. Moody, “Diffraction of stochastic point sets: explicitly computable examples,” Communications in Mathematical Physics, vol. 293, no. 3, pp. 611–660, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. J. Bellissard, D. J. L. Herrmann, and M. Zarrouati, “Hulls of aperiodic solids and gap labeling theorems,” in Directions in Mathematical Quasicrystals, vol. 13 of CRM Monograph Series, pp. 207–258, American Mathematical Society, Providence, RI, USA, 2000. View at Google Scholar · View at Zentralblatt MATH
  6. J. Kellendonk, “Noncommutative geometry of tilings and gap labelling,” Reviews in Mathematical Physics, vol. 7, no. 7, pp. 1133–1180, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. L. Sadun, “Tilings, tiling spaces and topology,” Philosophical Magazine, vol. 86, no. 6–8, pp. 875–881, 2006. View at Publisher · View at Google Scholar
  8. L. Sadun, Topology of Tiling Spaces, vol. 46 of University Lecture Series, American Mathematical Society, Providence, RI, USA, 2008.
  9. K.-P. Nischke and L. Danzer, “A construction of inflation rules based on n-fold symmetry,” Discrete & Computational Geometry, vol. 15, no. 2, pp. 221–236, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  10. J. G. Escudero, “Random tilings of compact Euclidean 3-manifolds,” Acta Crystallographica. Section A, vol. 63, no. 5, pp. 391–399, 2007. View at Publisher · View at Google Scholar · View at PubMed
  11. J. G. Escudero, “Random tilings of spherical 3-manifolds,” Journal of Geometry and Physics, vol. 58, no. 11, pp. 1451–1464, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. J. García-Escudero, “Grammars for icosahedral Danzer tilings,” Journal of Physics. A, vol. 28, no. 18, pp. 5207–5215, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. B. Grünbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman and Company, New York, NY, USA, 1987.
  14. C. Godrèche and J. M. Luck, “Quasiperiodicity and randomness in tilings of the plane,” Journal of Statistical Physics, vol. 55, no. 1-2, pp. 1–28, 1989. View at Publisher · View at Google Scholar
  15. P. Gummelt, “Generalized substitution rules: concepts and examples,” Zeitschrift fur Kristallographie, vol. 223, no. 11-12, pp. 805–808, 2008. View at Publisher · View at Google Scholar
  16. M. Baake and U. Grimm, “Kinematic diffraction is insufficient to distinguish order from disorder,” Physical Review B, vol. 79, no. 2, Article ID 020203, 2009. View at Publisher · View at Google Scholar
  17. J. E. Anderson and I. F. Putnam, “Topological invariants for substitution tilings and their associated C-algebras,” Ergodic Theory and Dynamical Systems, vol. 18, no. 3, pp. 509–537, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  18. N. Ormes, C. Radin, and L. Sadun, “A homeomorphism invariant for substitution tiling spaces,” Geometriae Dedicata, vol. 90, pp. 153–182, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. J. Kellendonk and I. F. Putnam, “Tilings, C-algebras, and K-theory,” in Directions in Mathematical Quasicrystals, vol. 13 of CRM Monograph Series, pp. 177–206, American Mathematical Society, Providence, RI, USA, 2000. View at Google Scholar
  20. M. Barge, B. Diamond, J. Hunton, and L. Sadun, “Cohomology of substitution tiling spaces,” Ergodic Theory and Dynamical Systems, vol. 30, no. 6, pp. 1607–1627, 2010. View at Publisher · View at Google Scholar
  21. J. G. Escudero, “Integer Cech cohomology of a class of n-dimensional substitutions,” Mathematical Methods in the Applied Sciences, vol. 34, no. 5, pp. 587–594, 2011. View at Publisher · View at Google Scholar
  22. J. Kellendonk and I. F. Putnam, “The Ruelle-Sullivan map for actions of Rn,” Mathematische Annalen, vol. 334, no. 3, pp. 693–711, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. M. Baake, P. Kramer, M. Schlottmann, and D. Zeidler, “Planar patterns with fivefold symmetry as sections of periodic structures in 4-space,” International Journal of Modern Physics B, vol. 4, no. 15-16, pp. 2217–2268, 1990. View at Publisher · View at Google Scholar
  24. F. Gähler, J. Hunton, and J. Kellendonk, “Torsion in Tiling Homology and Cohomology,” http://arxiv.org/abs/math-ph/0505048.
  25. C. Goodman-Strauss, “Matching rules and substitution tilings,” Annals of Mathematics. Second Series, vol. 147, no. 1, pp. 181–223, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet