An Algorithmic Approach to Tilings

  • A. Dress
  • O. Delgado Friedrichs
  • D. Huson
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 329)

Abstract

The obvious structuredness inherent to regular periodic tilings has been so self-evident that it took almost two and a half thousand years after the first complete classification and enumeration results had been obtained before people started to try to explicitly address this particular structuredness in terms of appropriate formal concepts capturing its essence.

Keywords

Euclidean Plane Hyperbolic Plane Geometric Realization Symbolic Level Symbolic Approach 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References

  1. [1]
    L. Bieberbach, Über die Bewegungsgruppen der Euklidischen Räume I, Math. Ann. 70 (1911), 297.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    L. Bieberbach, Über die Bewegungsgruppen der Euklidischen Räume II, Math. Ann. 72 (1912), 400.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    J. Bornhöft, Zum einfachen Zusammenhang von Kantenkoronen und einem sich daraus ergebenden iterativen Algorithmus zur (Re-)Konstruk- Hon planarer Pflasterungen, Master’s thesis, University of Bielefeld, 1994.Google Scholar
  4. [4]
    O. Delgado Friedrichs, A.W.M. Dress, A. Muller, and M.T. Pope, Poly-oxometalates: A class of compounds with remarkable topology, Molecular Engineering 3 (1993), 9 - 28.CrossRefGoogle Scholar
  5. [5]
    B.N. Delone, Teoriya planigonov, Ivz. Akad. Nauk SSSR, ser. Matem. 23 (1961), 365–386.MathSciNetGoogle Scholar
  6. [6]
    M.S. Delaney, Quasisymmetries of space group orbits, Match 9 (1980), 73–80.MathSciNetMATHGoogle Scholar
  7. [7]
    O. Delgado Friedrichs, Die automatische Konstruktion periodischer Pflasterungen, Master’s thesis, University of Bielefeld, 1990.Google Scholar
  8. [8]
    O. Delgado Friedrichs, Euclidicity criteria for three-dimensional branched triangulations, Ph.D. thesis, University of Bielefeld, 1994.MATHGoogle Scholar
  9. [9]
    A.W.M. Dress and D.H. Huson, Heaven and hell tilings, Revue Topologie Structurale 17 (1991), 25–42.MathSciNetMATHGoogle Scholar
  10. [10]
    O. Delgado Friedrichs and D.H. Huson, REPTILES, University of Bielefeld, 1992, Shareware macintosh-program.Google Scholar
  11. [11]
    A.W.M. Dress, D.H. Huson, and E. Molnár, The classification of face-transitive 3d-tilings, Acta Crystallographica A49 (1993), 806–817.Google Scholar
  12. [12]
    O. Delgado Friedrichs, D.H. Huson, and E. Zamorzaeva, The classification of 2-isohedral tilings of the plane, Geometriae Dedicata 42 (1992), 43–117.MathSciNetGoogle Scholar
  13. [13]
    A.W.M. Dress, On the classification and generation of two- and higher- dimensional regular patterns, Match 9 (1980), 81–100.MathSciNetMATHGoogle Scholar
  14. [14]
    A.W.M. Dress, Regular polytopes and equivariant tessellations from a combinatorial point of view, Algebraic Topology (Göttingen), SLN 1172, Göttingen, 1984, pp. 56–72.Google Scholar
  15. [15]
    A.W.M. Dress, Presentations of discrete groups, acting on simply connected manifolds, Adv. in Math. 63 (1987), 196–212.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    A.W.M. Dress and R. Scharlau, Zur Klassifikation aquivarianter Pflasterungen, Mitteilungen aus dem Math. Seminar Giessen 164 (1984), 83–136.MathSciNetGoogle Scholar
  17. [17]
    A.W.M. Dress and R. Scharlau, The 37 combinatorial types of minimal, non-transitive, equivariant tilings of the euclidean plane, Discrete Math. 60 (1986), 121–138.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    B. Grünbaum and G.C. Shephard, Tilings and patterns, W.H. Freeman and Company, New York, 1987.MATHGoogle Scholar
  19. [19]
    H. Heesch, Regulãres Parkettierungsproblemy, Westdeutscher Verlag, Köln-Opladen, 1968.Google Scholar
  20. [20]
    D.H. Huson, The generation and classification of tile-k-transitive tilings of the euclidean plane, the sphere and the hyperbolic plane, Geometriae Dedicata 47 (1993), 269–296.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    D.H. Huson, Tile-transitive partial tilings of the plane, Contributions to Geometry and Algebra 34 (1993), no. 1, 87–118.MathSciNetMATHGoogle Scholar
  22. [22]
    D.H. Huson, A four-color theorem for periodic tilings, Geometriae Dedicata 51 (1994), 47–61.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    Martin Schönert et al., GAP — Groups, Algorithms, and Programming, Lehrstuhl D für Mathematik, Rheinisch Westfalische Technische Hochschule, Aachen, Germany, third ed., 1993.Google Scholar
  24. [24]
    K. Westphal, Zur Konstruktion zweidimensionaler Pflasterungen in allen drei Geometrien, Master’s thesis, University of Bielefeld, 1991.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • A. Dress
    • 1
  • O. Delgado Friedrichs
    • 1
  • D. Huson
    • 1
  1. 1.Universität BielefeldGermany

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