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Reptilings and space-filling curves for acute triangles

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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.

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Acknowledgements

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

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Authors and Affiliations

  1. Technische Universität München, 80333, Munich, Germany

    Marinus Gottschau

  2. Technische Universiteit Eindhoven, 5600, Eindhoven, Netherlands

    Herman Haverkort

  3. Ludwig-Maximilians-Universität München, 80333, Munich, Germany

    Kilian Matzke

Authors
  1. Marinus Gottschau
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  2. Herman Haverkort
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  3. Kilian Matzke
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Correspondence to Kilian Matzke.

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Gottschau, M., Haverkort, H. & Matzke, K. Reptilings and space-filling curves for acute triangles. Discrete Comput Geom 60, 170–199 (2018). https://doi.org/10.1007/s00454-017-9953-0

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  • DOI: https://doi.org/10.1007/s00454-017-9953-0

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