Mixed labyrinth fractals

TL;DR

This paper introduces mixed labyrinth fractals, a generalization of labyrinth fractals that are not necessarily self-similar, and explores their geometric and topological properties, including path lengths, dimensions, and connections to Sierpinski carpets.

Contribution

It defines and studies mixed labyrinth fractals constructed via sequences of patterns, extending previous work on self-similar labyrinth fractals, and investigates their properties and related conjectures.

Findings

Mixed labyrinth fractals are dendrites.

Analyzed path lengths and box counting dimensions.

Established connections to generalized Sierpinski carpets.

Abstract

Labyrinth fractals are self-similar fractals that were introduced and studied in recent work by Cristea and Steinsky. In the present paper we define and study more general objects, called mixed labyrinth fractals, that are in general not self-similar and are constructed by using sequences of labyrinth patterns. We show that mixed labyrinth fractals are dendrites and study properties of the paths in the graphs associated to prefractals, and of arcs in the fractal, e.g., the path length and the box counting dimension and length of arcs. We also consider more general objects related to mixed labyrinth fractals, formulate two conjectures about arc lengths, and establish connections to recent results on generalised Sierpinski carpets.