### Abstract

An r-gentiling is a dissection of a shape into r≥ 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.

Original language | English |
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Pages (from-to) | 170-199 |

Number of pages | 30 |

Journal | Discrete and Computational Geometry |

Volume | 60 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jul 2018 |

### Keywords

- Meshing
- Reptile
- Space-filling curve
- Tessellation

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## Cite this

*Discrete and Computational Geometry*,

*60*(1), 170-199. https://doi.org/10.1007/s00454-017-9953-0