Gottschau, Marinus; Haverkort, Herman; Matzke, Kilian
(2018):
Reptilings and spacefilling curves for acute triangles.
In: Discrete & Computational Geometry, Vol. 60, No. 1: pp. 170199

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Abstract
An rgentiling is a dissection of a shape into parts that are all similar to the original shape. An rreptiling is an rgentiling 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 facecontinuous spacefilling curve if, at any level of recursion, the interior of the union of any set of consecutive parts is connectedthat is, with twodimensional shapes, consecutive parts must always meet along an edge. Most famously, the isosceles right triangle admits a 2reptiling, which can be used to describe the facecontinuous SierpiA"ski/Plya spacefilling curve;many other right triangles admit reptilings and gentilings that yield facecontinuous spacefilling curves as well. In this study we investigate which acute triangles admit nontrivial reptilings and gentilings, and whether these can form the basis for facecontinuous spacefilling curves. We derive several properties of reptilings and gentilings of acute (sometimes also obtuse) triangles, leading to the following conclusion: no facecontinuous spacefilling curve can be constructed on the basis of reptilings of acute triangles.