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Quasi-Periodicity in Islamic Geometric Design

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Architecture and Mathematics from Antiquity to the Future

Abstract

For nearly 150 years, scholars have analyzed the symmetries of Islamic ornamental designs, constituting the most highly developed chapter in cultural symmetry studies. Yet these studies hardly exhaust the mathematically significant properties of Islamic designs. Over the past 30 years, mathematicians have given increasing attention to “quasi-periodic” geometric structures (such as the Penrose tilings) which exhibit infinite repetition of their bounded subparts and crystallographically forbidden symmetries, occupying an important niche between random and highly ordered, periodic structures. This chapter provides a critical review of the literature and argues that the dualization of quasi-periodic tiling fragments from two designs from twelfth- and fifteenth-century Iran helps to inform their aesthetic complexity.

First published as: Peter Saltzman , “Quasi-Periodicity in Islamic Geometric Design”. Pp. 153–168 in Nexus VII: Architecture and Mathematics, Kim Williams, ed. Turin: Kim Williams Books, 2008.

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Notes

  1. 1.

    For a general introduction to the cultural applications of symmetry studies, see Crowe and Washburn (1991).

  2. 2.

    Good introductions to the symmetries of Islamic geometric designs include the delightful book by Fenoll Hach-Alí and Galindo (2003) and the more comprehensive work by Abas and Salman (1995).

  3. 3.

    Several works analyse the orbit structure of Islamic designs in this manner, including Grünbaum and Shepherd (1986); Abas and Salman (1995); Ostromoukhov (1998).

  4. 4.

    Discussions of this technique may be found in Hankin (1925, 1934); Wade (1976); Bonner (2003); Kaplan (2005).

  5. 5.

    An extensive discussion may be found in Bonner (2000).

  6. 6.

    See, for example, Grünbaum and Shepherd (1987: Chap. 1).

  7. 7.

    Good introductions to this subject may be found in Grünbaum and Shepherd (1987) and Senechal (1995).

  8. 8.

    An interesting discussion of Birkhoff measures relevant to these remarks is found in Rigau et al. (2007). A study of the Kolmogorov complexity of finite subsets of tilings of the plane is found in Durand et al. (2008).

  9. 9.

    Similar views are expressed in Arnheim (1971).

  10. 10.

    Such designs have been produced by Rigby (2006).

  11. 11.

    Aspects of the M2 tilings later appeared in high resolution transmission electron microscopy of aluminium cobalt nickel alloys; see Cervellino et al. (2002).

  12. 12.

    For further discussion of inflation tilings, see Senechal (1995: Chap. 5).

  13. 13.

    The Gunbad-i Kabud has deteriorated, and thus the original design is obscured in parts. Lu and Steinhardt (2007a: Fig. S6) had pointed out that the lower portion of the original reconstruction given in Makovicky (1992) was incorrect. Emil Makovicky has carefully reconstructed the design based on his inspection of the building (Makovicky 2009).

  14. 14.

    Although not addressed in Bonner (2003), the subdivisions he uses are in fact part of a set of inflation rules that can also establish the quasi-periodicity of a segment of this design. Those inflation rules, however, require a subdivision rule for the narrow rhomb and one for an extra pentagon, and require more complex matching rules than those used by Lu and Steinhardt.

  15. 15.

    The large scale dual design from Darb-i Imam appears in multiple guises at many other sites with explicit periodic repetition, though it does not do so in the spandrel studied by Lu and Steinhardt. See Arik and Sancak (2007) and Cromwell (2009).

  16. 16.

    Castera (1999) has claimed that projections of three-dimensional Islamic “Muqarnas” are quasi-periodic, although Makovicky and Fenoll Hach-Ali (2001) have reached contrary conclusions.

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Acknowledgments

I would like to thank Emil Makovicky , Peter Lu and Peter Cromwell for sharing their work with me and for helpful comments on an earlier draft of this text.

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  1. 9003, Berkeley, CA, 94709, USA

    Peter Saltzman

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  1. Peter Saltzman
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Correspondence to Peter Saltzman .

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  1. Kim Williams Books, Torino, Italy

    Kim Williams

  2. School of Architecture and Built Environment, The University of Newcastle, Callaghan, Australia

    Michael J. Ostwald

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Saltzman, P. (2015). Quasi-Periodicity in Islamic Geometric Design. In: Williams, K., Ostwald, M. (eds) Architecture and Mathematics from Antiquity to the Future. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-00137-1_39

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