  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.
