Advertisement

Discrete & Computational Geometry

, Volume 60, Issue 1, pp 170–199 | Cite as

Reptilings and space-filling curves for acute triangles

  • Marinus Gottschau
  • Herman Haverkort
  • Kilian Matzke
Article
  • 119 Downloads

Abstract

An r-gentiling is a dissection of a shape into \(r \ge 2\) parts that are all similar to the original shape. An r-reptiling is an r-gentiling of which all parts are mutually congruent. By applying gentilings recursively, together with a rule that defines an order on the parts, one may obtain an order in which to traverse all points within the original shape. We say such a traversal is a face-continuous space-filling curve if, at any level of recursion, the interior of the union of any set of consecutive parts is connected—that is, with two-dimensional shapes, consecutive parts must always meet along an edge. Most famously, the isosceles right triangle admits a 2-reptiling, which can be used to describe the face-continuous Sierpiński/Pólya space-filling curve; many other right triangles admit reptilings and gentilings that yield face-continuous space-filling curves as well. In this study we investigate which acute triangles admit non-trivial reptilings and gentilings, and whether these can form the basis for face-continuous space-filling curves. We derive several properties of reptilings and gentilings of acute (sometimes also obtuse) triangles, leading to the following conclusion: no face-continuous space-filling curve can be constructed on the basis of reptilings of acute triangles.

Keywords

Reptile Tessellation Space-filling curve Meshing 

Notes

Acknowledgements

We thank Dirk Gerrits for his help in obtaining our first proofs of Theorem 5.6.

References

  1. 1.
    Bader, M., Zenger, C.: Efficient storage and processing of adaptive triangular grids using Sierpiński curves. In: Proceedings of the 6th International Conference on Computational Science (ICCS’06). Lecture Notes in Computer Science, vol. 3991, pp. 673–680. Springer, Berlin (2006)Google Scholar
  2. 2.
    Freese, R.W., Miller, A.K., Usiskin, Z.: Can every triangle be divided into \(n\) triangles similar to it? Am. Math. Mon. 77(8), 867–869 (1990)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Golomb, S.W.: Replicating figures in the plane. Math. Gaz. 48(366), 403–412 (1964)CrossRefzbMATHGoogle Scholar
  4. 4.
    Haverkort, H., McGranaghan, M., Toma, L.: An edge quadtree for external memory. In: Proceedings of the 12th International Symposium on Experimental Algorithms (SEA’13). Lecture Notes in Computer Science, vol. 7933, pp. 115–126. Springer, Berlin (2013)Google Scholar
  5. 5.
    Hilbert, D.: Ueber die stetige Abbildung einer Linie auf ein Flächenstück. Math. Ann. 38(3), 459–460 (1891)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kaiser, H.: Selbstähnliche Dreieckszerlegungen. Friedrich-Schiller-Universität, Jena (1990)Google Scholar
  7. 7.
    Kamel, I., Faloutsos, C.: On packing R-trees. In: Bhargava, B., et al. (eds.) Proceedings of the 2nd International Conference on Information and Knowledge Management (CIKM’93), pp. 490–499. ACM, New York (1993)Google Scholar
  8. 8.
    Lebesgue, H.: Leçons sur l’intégration et la recherche des fonctions primitives, pp. 44–45. Gauthier-Villars, Paris (1904)zbMATHGoogle Scholar
  9. 9.
    Morton, G.M.: A Computer Oriented Geodetic Data Base and a New Technique in File Sequencing. International Business Machines Corporation, Ottawa (1966)Google Scholar
  10. 10.
    Peano, G.: Sur une courbe, qui remplit toute une aire plane. Math. Ann. 36(1), 157–160 (1890)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Platzman, L.K., Bartholdi III, J.J.: Spacefilling curves and the planar travelling salesman problem. J. ACM 36(4), 719–737 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Pólya, G.: Über eine Peanosche Kurve. Bull. Int. Acad. Sci. Cracovie A 1913, 305–313 (1913)Google Scholar
  13. 13.
    Sagan, H.: Space-Filling Curves. Universitext. Springer, New York (1994)CrossRefzbMATHGoogle Scholar
  14. 14.
    Sierpiński, W.: Oeuvres Choisies, vol. II, pp. 52–66. PWN, Warszawa (1975)Google Scholar
  15. 15.
    Snover, S.L., Waiveris, C., Williams, J.K.: Rep-tiling for triangles. Discrete Math. 91(2), 193–200 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Technische Universität MünchenMunichGermany
  2. 2.Technische Universiteit EindhovenEindhovenNetherlands
  3. 3.Ludwig-Maximilians-Universität MünchenMunichGermany

Personalised recommendations