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Some Problems on Plane Tilings

  • Branko Grünbaum
  • G. C. Shephard

Abstract

Although the art of tiling is as old as human history, the science of tiling seems to have been curiously neglected until recent times.

Keywords

Extension Theorem Topological Disk Periodic Tiling Plane Tiling Convex Pentagon 
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 and Further Reading

  1. Several collections of Escher’s works have been published; the most extensive is The World of M. C. Escher (Abrams, New York, 1971). A very interesting account of Escher and his tilings is given in B. Ernst’s book The Magic Mirror of M. C. Escher (Random House, New York, 1976). A discussion of the tilings to be found in the Alhambra appears in E. Müller, “Gruppentheoretische Ornamente aus der Alhambra in Grenada” (ETH dissertation, Zürich, 1944). Kepler’s book Harrnonice Mundi, originally published in Linz in 1619, has been reprinted in Kepler’s Complete Works, edited by M. Caspar (Gesammelte Werke, Band VI, Beck, München, 1940 and by Culture et Civilisation, Bruxelles, 1968). These texts are in Latin; a German translation, Weltharmonik, by M. Caspar has also been published (Oldenbourg, München, 1967).Google Scholar
  2. 1.
    The three types of hexagons that admit tilings of the plane were determined by K. Reinhardt in his thesis “Über die Zerlegung der Ebene in Polygone”, Frankfurt University, 1918 (Noske, Leipzig, 1918). Kershner’s paper is “On paving the plane”, American Mathematical Monthly 75 (1968), 839–844; our list of pentagons is taken from D. Schattschneider, “Tiling the plane with congruent pentagons”, Mathematics Magazine 51 (1978), 29–44.CrossRefGoogle Scholar
  3. 1a.
    The three types of hexagons that admit tilings of the plane were determined by K. Reinhardt in his thesis “Über die Zerlegung der Ebene in Polygone”, Frankfurt University, 1918 (Noske, Leipzig, 1918). Kershner’s paper is “On paving the plane”, American Mathematical Monthly 75 (1968), 839–844; our list of pentagons is taken from D. Schattschneider, “Tiling the plane with congruent pentagons”, Mathematics Magazine 51 (1978), 29–44. The announcement by M. D. Hirschhorn and D. C. Hunt that the list of equilateral pentagons is complete also appears in the Mathematics Magazine 51 (1978), p. 312.Google Scholar
  4. 2.
    The subject of k-morphic tilings is considered in the authors’ “Patch-determined tilings”, Mathematical Gazette 61 (1977), 31–38, and Harborth’s example in “Prescribed numbers of tiles and tilings”, Mathematical Gazette 61 (1977), 296–299.Google Scholar
  5. 3.
    Hilbert’s famous problems were printed (English translation) in “Mathematical Problems”, Bulletin of the American Mathematical Society 8 (1902), 437–479 and reprinted in “Mathematical Developments Arising from Hilbert Problems”, Proc. Sympos. Pure Math., Vol. 28, (American Math. Soc, Providence, R.I., 1976). The book by Heesch and Kienzle is Flächenschluss (Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963). A recent treatment of isohedral tilings is the authors’ “The eighty-one types of isohedral tilings in the plane”, Math. Proc. Cambridge Philos. Soc. 82 (1977), 177–196.Google Scholar
  6. 4.
    A proof of the Extension Theorem will appear in the authors’ book mentioned below. Heesch’s problem appears in his book Reguläres Parkettierungsproblem (Westdeutscher Verlag, Köln-Opladen, 1968).Google Scholar
  7. 5.
    Accounts of aperiodic tiles appear in R. M. Robinson’s article “Undecidability and non-periodicity of tilings of the plane”, Inventiones Math. 12 (1971), 177–209.CrossRefGoogle Scholar
  8. 5a.
    R. Penrose’s papers “The role of aesthetics in pure and applied mathematical research”, Bull. Inst. Math. Appl. 10 (1974), 266–271, and “Pentaplexity”, Eureka 39 (1978), 16–22. The most up-to-date account is Martin Gardner’s article cited in the text, and a more detailed exposition will appear in the author’s forthcoming book.Google Scholar
  9. 5b.
    The connection between aperiodicity and the tiling problem is discussed in Robinson’s paper quoted above, and also in H. Wang, “Proving theorems by pattern recognition II”, Bell System Techn. Journal 40 (1961), 1–42.Google Scholar
  10. 6.
    Voderberg’s papers are “Zur Zerlegung der Umgebung eines ebenen Bereiches in kongruente”, J.-Ber. Deutsch. Math.-Verein. 46 (1936), 229–231, and “Zur Zerlegung der Ebene in kongruente Bereiche in Form einer Spirale”, ibid. 47 (1937), 159–160. Goldberg’s explanation of the structure of spiral tilings appears in “Central tessellations”, Scripta Math. 21 (1955), 253–260. A short article “Spiral tilings and versatiles” by the authors appeared in Mathematics Teaching 88 (1979), 50–51.Google Scholar

Copyright information

© Wadsworth International 1981

Authors and Affiliations

  • Branko Grünbaum
    • 1
  • G. C. Shephard
    • 2
  1. 1.University of WashingtonUSA
  2. 2.University of East AngliaUK

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