Abstract
Labyrinth fractals are self-similar dendrites in the unit square that are defined with the help of a labyrinth set or a labyrinth pattern. In the case when the fractal is generated by a horizontally and vertically blocked pattern, the arc between any two points in the fractal has infinite length (Cristea and Steinsky in Geom Dedicata 141(1):1–17, 2009; Proc Edinb Math Soc 54(2):329–344, 2011). In the case of mixed labyrinth fractals a sequence of labyrinth patterns is used in order to construct the dendrite. In the present article we focus on the length of the arcs between points of mixed labyrinth fractals. We show that, depending on the choice of the patterns in the sequence, both situations can occur: the arc between any two points of the fractal has finite length, or the arc between any two points of the fractal has infinite length. This is in stark contrast to the self-similar case.
Highlights
Labyrinth fractals are fractal dendrites in the plane, that can be viewed as a special family of Sierpinski carpets
In the present article we show that in the case of mixed labyrinth fractals the situation is much more complex: on the one hand, one can find sequences of blocked labyrinth patterns that generate labyrinth fractals where the arc between any two points in the fractal is finite, and on the other hand one can find sequences of blocked labyrinth patterns whose resulting labyrinth fractal has the property that the arc between any two points of the fractal has infinite length
The results in this article provide ideas and modalities for constructing such fractal dendrites with desired properties regarding the lengths of arcs beween points in the fractal, that could serve as models, e.g., in the context of particle transport, nanostructures, image processing
Summary
Labyrinth fractals are fractal dendrites in the plane, that can be viewed as a special family of Sierpinski carpets. The results in this article provide ideas and modalities for constructing such fractal dendrites with desired properties regarding the lengths of arcs beween points in the fractal, that could serve as models, e.g., in the context of particle transport, nanostructures, image processing. We note that random Koch curves, i.e., objects that are related, e.g., to arcs between certain points (exit points) in labyrinth fractals, are studied with respect to random walks by theoretical physicists in connection with diffusion processes, e.g., [13]. In this context we mention diffusion processes of water in biological tissues. There are many more available examples that support the idea that labyrinth fractals, whether mixed or self-similar, are mathematical objects worth understanding with respect to their topological and geometrical properties, with benefits both in mathematics and in other sciences
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