Reptilings and space-filling curves for acute triangles
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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.
KeywordsReptile Tessellation Space-filling curve Meshing
We thank Dirk Gerrits for his help in obtaining our first proofs of Theorem 5.6.
- 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
- 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
- 6.Kaiser, H.: Selbstähnliche Dreieckszerlegungen. Friedrich-Schiller-Universität, Jena (1990)Google Scholar
- 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
- 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
- 12.Pólya, G.: Über eine Peanosche Kurve. Bull. Int. Acad. Sci. Cracovie A 1913, 305–313 (1913)Google Scholar
- 14.Sierpiński, W.: Oeuvres Choisies, vol. II, pp. 52–66. PWN, Warszawa (1975)Google Scholar