Triangular labyrinth fractals

Abstract

We define and study a class of fractal dendrites called triangular labyrinth fractals. For the construction, we use triangular labyrinth patterns systems that consist of two triangular patterns: a white and a yellow one. Correspondingly, we have two fractals: a white and a yellow one. The fractals studied here are self-similar, and fit into the framework of graph directed constructions. The main results consist in showing how special families of triangular labyrinth patterns systems, defined based on some shape features, can generate exactly three types of dendrites: labyrinth fractals where are all nontrivial arcs have infinite length, fractals where all nontrivial arcs have finite length, and fractals where the only arcs of finite length are line segments parallel to a certain direction. We also study the existence of tangents to arcs. The paper is inspired by research done on labyrinth fractals in the unit square that have been studied during the last decade. In the triangular case, due to the geometry of traingular shapes, some new techniques and ideas are necessary in order to obtain the results.