On the length of arcs in labyrinth fractals

TL;DR

This paper investigates the lengths of arcs in mixed labyrinth fractals, revealing that depending on the pattern sequence, arcs can be either finite or infinite, contrasting with the self-similar case where arcs are always infinite.

Contribution

It demonstrates that in mixed labyrinth fractals, the arc length between points can be finite or infinite based on pattern choices, unlike self-similar fractals.

Findings

Arc lengths can be finite or infinite depending on pattern sequences.

Contrasts with self-similar labyrinth fractals where arcs are always infinite.

Provides conditions for finiteness or infiniteness of arc lengths.

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\&Steinsky 2009,Cristea\&Steinsky 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.