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


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.


Reptile Tessellation Space-filling curve Meshing 



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


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Authors and Affiliations

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

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