# Reptilings and space-filling curves for acute triangles

Marinus Gottschau, Herman Haverkort, Kilian Matzke (Corresponding author)

### 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 170-199 30 Discrete and Computational Geometry 60 1 10.1007/s00454-017-9953-0 Gepubliceerd - 1 jul 2018

### Vingerafdruk

Acute triangle
Space-filling Curves
Dissection
Face
Right-angled triangle
Consecutive
Obtuse triangle
Isosceles triangle
Congruent
Recursion
Acute
Interior
Union

### Citeer dit

Gottschau, Marinus ; Haverkort, Herman ; Matzke, Kilian. / Reptilings and space-filling curves for acute triangles. In: Discrete and Computational Geometry. 2018 ; Vol. 60, Nr. 1. blz. 170-199
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Gottschau, M, Haverkort, H & Matzke, K 2018, 'Reptilings and space-filling curves for acute triangles' 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).

In: Discrete and Computational Geometry, Vol. 60, Nr. 1, 01.07.2018, blz. 170-199.

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Gottschau M, Haverkort H, Matzke K. Reptilings and space-filling curves for acute triangles. Discrete and Computational Geometry. 2018 jul 1;60(1):170-199. Beschikbaar vanaf, DOI: 10.1007/s00454-017-9953-0