ON MIXED TRIANGULAR LABYRINTHIC FRACTALS

Abstract

We introduce and study mixed triangular labyrinthic fractals, which can be seen as an extension of (generalized) Sierpiński gaskets. This is a new class of fractal dendrites that generalize the self-similar triangular labyrinth fractals studied recently by the authors. A mixed triangular labyrinthic fractal is defined by a triangular labyrinthic pattern and an infinite sequence of triangular labyrinth patterns systems, whereas one triangular labyrinth patterns system is sufficient for generating a self-similar triangular labyrinth fractal. Moreover, a labyrinthic pattern is more general than a labyrinth pattern, and the idea is that the use of many different patterns provides objects with a “richer” structure. After proving that these non-self-similar fractals are dendrites, we study the growth of path lengths in graphs associated with iterations. We prove geometric properties of the fractals, some of which have rather “local” character and on the other hand occur at sites distributed “all over” the fractals.