  An $r$-gentiling is a dissection of a shape into $r \geq 2$ parts which 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, consecutive parts must always meet along an edge. Most
famously, the isosceles right triangle admits a 2-reptiling, which forms the
basis of the face-continuous Sierpinski 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 what 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.

Combinatorics;Metric Geometry