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

AN - SCOPUS:85037695431

SN - 0179-5376

VL - 60

SP - 170

EP - 199

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 1

ER -