### Uittreksel

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.

Taal | Engels |
---|---|

Pagina's | 170-199 |

Aantal pagina's | 30 |

Tijdschrift | Discrete and Computational Geometry |

Volume | 60 |

Nummer van het tijdschrift | 1 |

DOI's | |

Status | Gepubliceerd - 1 jul 2018 |

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### Citeer dit

*Discrete and Computational Geometry*,

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

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*Discrete and Computational Geometry*, vol. 60, nr. 1, blz. 170-199. DOI: 10.1007/s00454-017-9953-0

**Reptilings and space-filling curves for acute triangles.** / Gottschau, Marinus; Haverkort, Herman; Matzke, Kilian (Corresponding author).

Onderzoeksoutput: Bijdrage aan tijdschrift › Tijdschriftartikel › Academic › peer review

TY - JOUR

T1 - Reptilings and space-filling curves for acute triangles

AU - Gottschau,Marinus

AU - Haverkort,Herman

AU - Matzke,Kilian

PY - 2018/7/1

Y1 - 2018/7/1

N2 - 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.

AB - 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.

KW - Meshing

KW - Reptile

KW - Space-filling curve

KW - Tessellation

UR - http://www.scopus.com/inward/record.url?scp=85037695431&partnerID=8YFLogxK

U2 - 10.1007/s00454-017-9953-0

DO - 10.1007/s00454-017-9953-0

M3 - Article

VL - 60

SP - 170

EP - 199

JO - Discrete and Computational Geometry

T2 - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 1

ER -