The Search for Quasi-Periodicity in Islamic 5-fold Ornament

Article

Notes

Acknowledgments

I would like to thank Paul Steinhardt for clarifying some statements in [17] and Peter Saltzman for sharing a draft of his article [29]. I am also very grateful to the following people for their critical reading of an early draft of this article and for suggesting improvements: Helmer Aslaksen, Elisabetta Beltrami, Jean-Marc Castéra, Dirk Frettlöh, Chaim␣Goodman-Strauss, Emil Makovicky, John Rigby, Joshua Socolar, and John Sullivan.

Bibliography

  1. 1.
    M. Arik and M. Sancak, ‹Turkish–Islamic art and Penrose tilings’, Balkan Physics Letters 15 (1 Jul 2007) 1–12.Google Scholar
  2. 2.
    J. Bonner, ‹Three traditions of self-similarity in fourteenth and fifteenth century Islamic geometric ornament’, Proc. ISAMA/Bridges: Mathematical Connections in Art, Music and Science, (Granada, 2003), eds. R. Sarhangi and N. Friedman, 2003, pp. 1–12.Google Scholar
  3. 3.
    J. Bonner, Islamic Geometric Patterns: Their Historical Development and Traditional Methods of Derivation, unpublished manuscript.Google Scholar
  4. 4.
    J. Bourgoin, Les Eléments de l’Art Arabe: Le Trait des Entrelacs, Firmin-Didot, 1879. Plates reprinted in Arabic Geometric Pattern and Design, Dover Publications, 1973.Google Scholar
  5. 5.
    J.-M. Castéra, Arabesques: Art Décoratif au Maroc, ACR Edition, 1996.Google Scholar
  6. 6.
    J.-M. Castéra, ‹Zellijs, muqarnas and quasicrystals’, Proc. ISAMA, (San Sebastian, 1999), eds. N. Friedman and J. Barrallo, 1999, pp. 99–104.Google Scholar
  7. 7.
    G. M. Fleurent, ‹Pentagon and decagon designs in Islamic art’, Fivefold Symmetry, ed. I. Hargittai, World Scientific, 1992, pp.␣263–281.Google Scholar
  8. 8.
    B. Grünbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman, 1987.Google Scholar
  9. 9.
    E. H. Hankin, ‹On some discoveries of the methods of design employed in Mohammedan art’, J. Society of Arts 53 (1905) 461–477.Google Scholar
  10. 10.
    E. H. Hankin, The Drawing of Geometric Patterns in Saracenic Art, Memoirs of the Archaeological Society of India, no 15, Government of India, 1925.Google Scholar
  11. 11.
    E. H. Hankin, ‹Examples of methods of drawing geometrical arabesque patterns’, Math. Gazette 12 (1925) 370–373.CrossRefGoogle Scholar
  12. 12.
    E. H. Hankin, ‹Some difficult Saracenic designs II’, Math. Gazette 18 (1934) 165–168.CrossRefGoogle Scholar
  13. 13.
    E. H. Hankin, ‹Some difficult Saracenic designs III’, Math. Gazette 20 (1936) 318–319.CrossRefGoogle Scholar
  14. 14.
    C. S. Kaplan, ‹Computer generated Islamic star patterns’, Proc. Bridges: Mathematical Connections in Art, Music and Science, (Kansas, 2000), ed. R. Sarhangi, 2000, pp. 105–112.Google Scholar
  15. 15.
    C. S. Kaplan, ‹Islamic star patterns from polygons in contact’, Graphics Interface 2005, ACM International Conference Proceeding Series 112, 2005, pp. 177–186.Google Scholar
  16. 16.
    A. J. Lee, ‹Islamic star patterns’, Muqarnas IV: An Annual on Islamic Art and Architecture, ed. O. Grabar, Leiden, 1987, pp.␣182–197.Google Scholar
  17. 17.
    P. J. Lu and P. J. Steinhardt, ‹Decagonal and quasi-crystalline tilings in medieval Islamic architecture’, Science 315 (23 Feb 2007) 1106–1110.Google Scholar
  18. 18.
    P. J. Lu and P. J. Steinhardt, ‹Response to Comment on “Decagonal and quasi-crystalline tilings in medieval Islamic architecture”, Science 318 (30 Nov 2007) 1383.Google Scholar
  19. 19.
    F. Lunnon and P. Pleasants, ‹Quasicrystallographic tilings’, J.␣Math. Pures et Appliqués 66 (1987) 217–263.MATHMathSciNetGoogle Scholar
  20. 20.
    E. Makovicky, ‹800-year old pentagonal tiling from Maragha, Iran, and the new varieties of aperiodic tiling it inspired’, Fivefold Symmetry, ed. I. Hargittai, World Scientific, 1992, pp. 67–86.Google Scholar
  21. 21.
    E. Makovicky, ‹Comment on “Decagonal and quasi-crystalline tilings in medieval Islamic architecture”, Science 318 (30 Nov 2007) 1383.Google Scholar
  22. 22.
    E. Makovicky and P. Fenoll Hach-Alí, ‹Mirador de Lindaraja: Islamic ornamental patterns based on quasi-periodic octagonal lattices in Alhambra, Granada, and Alcazar, Sevilla, Spain’, Boletín Sociedad Española Mineralogía 19 (1996) 1–26.Google Scholar
  23. 23.
    E. Makovicky and P. Fenoll Hach-Alí, ‹The stalactite dome of the Sala de Dos Hermanas—an octagonal tiling?’, Boletín Sociedad Española Mineralogía 24 (2001) 1–21.Google Scholar
  24. 24.
    E. Makovicky, F. Rull Pérez and P. Fenoll Hach-Alí, ‹Decagonal patterns in the Islamic ornamental art of Spain and Morocco’, Boletín Sociedad Española Mineralogía 21 (1998) 107–127.Google Scholar
  25. 25.
    G. Necipoğlu, The Topkapi Scroll: Geometry and Ornament in Islamic Architecture, Getty Center Publication, 1995.Google Scholar
  26. 26.
    J. Rigby, ‹A Turkish interlacing pattern and the golden ratio’, Mathematics in School 34 no 1 (2005) 16–24.Google Scholar
  27. 27.
    J. Rigby, ‹Creating Penrose-type Islamic interlacing patterns’, Proc. Bridges: Mathematical Connections in Art, Music and Science, (London, 2006), eds. R. Sarhangi and J. Sharp, 2006, pp. 41–48.Google Scholar
  28. 28.
    F. Rull Pérez, ‹La noción de cuasi-cristal a través de los mosaicos árabes’, Boletín Sociedad Española Mineralogía 10 (1987) 291–298.Google Scholar
  29. 29.
    P. W. Saltzman, ‹Quasi-periodicity in Islamic ornamental design’, Nexus VII: Architecture and Mathematics, ed. K. Williams, 2008, pp. 153–168.Google Scholar
  30. 30.
    M. Senechal, Quasicrystals and Geometry, Cambridge Univ. Press, 1995.Google Scholar
  31. 31.
    M. Senechal and J. Taylor, ‹Quasicrystals: The view from Les Houches’, Math. Intelligencer 12 no 2 (1990) 54–64.MATHMathSciNetCrossRefGoogle Scholar

Internet Resources

  1. 32.
    ArchNet. Library of digital images of Islamic architecture, http://archnet.org/library/images/
  2. 33.
    E. Harriss and D. Frettlöh, Tilings Encyclopedia, http://tilings.math.uni-bielefeld.de/
  3. 34.
    C. S. Kaplan, taprats, computer-generated Islamic star patterns, http://www.cgl.uwaterloo.ca/~csk/washington/taprats/
  4. 35.
    P. J. Lu and P. J. Steinhardt, Supporting online material for [17], http://www.sciencemag.org/cgi/content/full/315/5815/1106/DC1
  5. 36.
    D. Wade, Pattern in Islamic Art: The Wade Photo-Archive, http://www.patterninislamicart.com/

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Pure Mathematics Division, Mathematical Sciences Building University of LiverpoolLiverpool England

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