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Coding and generating complex periodic patterns

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Abstract

This paper presents a method of coding and generating infinite classes of comlex periodic geometric pattern found in various cultures. The method uses a pattern code for the known symmetry groups and is based on the infinite classes of subdivisions of the fundamental regions of each symmetry group. The subdivisions provide a skeletal grid from which selected point, line or polygonal sequences can be extracted to produce an infinite variety of geometric patternswithin a symmetry group. Surprisingly, many complex architectural patterns from different periods and cultures, notably those from historic Islamic buildings, and Chinese window lattices, can be systematically obtained by this method. The pattern-codes provide a basis for generation of simple and complex geometric patterns, and their transformation from one to another. Some directions for systematic transformations based on meta-structural principles are outlined.

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References

  1. Ardalan N, Bakhtiar L (1973) Sense of unity. Univ Chicago

  2. Dye DS (1937) A grammar of chinese lattices. Harvard Univ

  3. Merne JG (1984) A handbook of celtic ornament. Mercier Pr (Ireland)

  4. Critchlow K (1980) Time stand still. G. Fraser

  5. Acland JH (1972) Medieval structure: the gothic vault. Univ Toronto

  6. Gaudi Antonio (1852–1926) Fundacion Caja de Pensiones, Madrid 1985

  7. Fuller RB, Marks R (1973) The dymaxion world of buckmunster Fuller. Anchor

  8. Le Ricolais R (1973) Survey of works, structural research 1935–71. Zodiac 22

  9. Nervi PL (1965) Aesthetics and Technology in Building. Harvard

  10. Mengeringhausen M (1975) Raumfachwerke aus Knoten und Staben. Bauverlag Berlin Wiesbaden

  11. Wachsman K (1961) Turning point in building. Reinhold

  12. See, for example, the architectural projects of Baer, Baracs, Burt, Emmerich, Friedman, Hecker, Neumann, Pearce, Safdie, Tyng, and others

  13. Jones O (1856) Grammar of ornament. London

  14. Stevens P (1980) A Handbook of patterns. M.I.T.

  15. Grunbaum B, Shaphard (1986) Tilings and patterns. Freemau

  16. Dress AWM (1984) Zum Problem der regelmässigen Flächenaufteilung. Univ Bielefeld

  17. Loeb A (1976) Space structures. Addison-Wesley

  18. Bourgoin J (1879) Les elements de l'art arabe: le trait des entrelacs. Firmin-Didot, Paris (Dover reprint 1973)

    Google Scholar 

  19. Hankin EH (1925) The drawing of geometric patterns in saracenic art. Memoirs of the Archaeological Survey of India, No 15, Calcutta

  20. Maldonado BP (1975/76) El arte hispano-musulman en su decoracion geometrica. Instituto Hispano-Arabe de Cultura, Madrid

    Google Scholar 

  21. Critchlow K (1976) Islamic patterns. Schoken

  22. Albarn K, Smith JM, Steele S, Walker D (1974) The language of pattern. Thames and Hudson

  23. El-Said I, Parman A (1976) Geometric concepts in islamic art. World of Islam Festival Publ

  24. Lalvani H (1982) Coding and generating islamic patterns. National Institute of Design, Ahmedabad, p 30

  25. Lalvani H (1982) Pattern-regeneration. In: Dashi S (ed) Impulse to Adorn. Marg Pub Bombay, p. 123–136

    Google Scholar 

  26. Siromoney G, Siromoney R, Krithivasan K (1974) Array grammars and kolam, Comp Graph Image Proc 3: 63–82

    Google Scholar 

  27. Lalvani H (1981) Multi-dimensional periodic arrangements of transforming space structures. PhD Thesis Univ Pennsylvania

  28. Lalvani H (1982) Structures on hyper-structures. Lalvani NY

    Google Scholar 

  29. Locher JL (ed) (1982) M.C. Escher, his life and complete graphic work. Abrams

  30. For W. Huff's work, see D. Hofstadter's article in Sci Am (July 1983)

  31. Lalvani H (1977) Transpolyhedra. Lalvani NY

    Google Scholar 

  32. Hanrahan P, Lalvani H (1984) (unpublished), see [45]

  33. Senechal M (1986) Escher designs on surfaces. In: M.C. Escher: art and science. Coexter HSM, Emmer M, Penrose R, Teuber ML (eds) North-Holland, 1986.

  34. Buerger M (1968) Elementary crystallography. M.I.T.

  35. Shubnikov AV, Koptsik VA (1974) Symmetry in science and art. Plenum

  36. Loeb A (1978) Color and symmetry. Krieger

  37. Henry, NFM, Londsdale K (1982) International tables for X-ray crystallography, vol 1. Kynock Press, Birmingham

    Google Scholar 

  38. Hickethier A (1969) Color mixing by numbers. Van Nostrand Reinhold

    Google Scholar 

  39. Lalvani H (1981) First displayed at the exhibit ‘Structures on Hyper-Structures’, School of Architecture. Pratt Institute (September 1981)

  40. De Bruijn (1986) Personal communication

  41. Lalvani H (1989) Morphological aspects of space structures. In: H. Nooshin (in press) (ed) Space Structures, Theory and Practice. Multi-Science

  42. Coxeter HSM (1940) Regular and semi-regular polytopes. Math Z 46:380–407

    Google Scholar 

  43. Burt M (1982) The wandering vertex method. Structural Topology 6

  44. Fuller RB (1979) Synergetics 2. McMillan

  45. Hanrahan P, Lalvani H, McDermott R (1984) Sketches of polyhedra transformations. A computer-animation video, 12 min color (Computer Graphics Laboratory, NYIT)

  46. Fuller RB (1975) Synergetics. McMillan

  47. Goldberg M (1937) Tohoku Math J 43:104–108

    Google Scholar 

  48. Dye DS (1974) Chinese lattice designs. Dover (reprint of [2])

  49. Dye DS (1974) Op. cit. Chinese lattice designs. Dover (reprint of [2]), p. 395, b

  50. Dye DS (1974) Op. cit. Chinese lattice designs. Dover (reprint of [2]), p 380

  51. Dye DS (1974) Op. cit. Chinese lattice designs. Dover (reprint of [2]), p 177

  52. Dye DS (1974) Op. cit. Chinese lattice designs. Dover (reprint of [2]), p 411, b

  53. Dye DS (1974) Op. cit. Chinese lattice designs. Dover (reprint of [2]), p 392, n

  54. Dye DS (1974) Op. cit. Chinese lattice designs. Dover (reprint of [2]), p 182

  55. Dye DS (1974) Op. cit. Chinese lattice designs. Dover (reprint of [2]), p 215

  56. Dye DS (1974) Op. cit. Chinese lattice designs. Dover (reprint of [2]), p 141

  57. Dye DS (1974) Op. cit. Chinese lattice designs. Dover (reprint of [2]), p 55

  58. Dye DS (1974) Op. cit. Chinese lattice designs. Dover (reprint of [2]), p 79

  59. Dye DS (1974) Op. cit. Chinese lattice designs. Dover (reprint of [2]), p 189, K 5b

  60. Dye DS (1974) Op. cit. Chinese lattice designs. Dover (reprint of [2]), p 193, K 10b

  61. Term first suggested in Lalvani H (1978) Meta-structures in architecture National Institute for Architectural Education, Faculty Grant Report; adapted from G Bateson's usage of “meta-pattern”

  62. Term suggested by Goranson HT (personal communication, 1987); usage differs from A. Tyng's earlier usage (personal communication, 1973)

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Lalvani, H. Coding and generating complex periodic patterns. The Visual Computer 5, 180–202 (1989). https://doi.org/10.1007/BF02153749

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