On the dimension of arcs in mixed labyrinth fractals

TL;DR

This paper explores the construction of mixed labyrinth fractals with arcs of varying box counting dimensions, including those with dimension 2, and extends these results to more general settings.

Contribution

It introduces a method to construct mixed labyrinth fractals with arcs of any box counting dimension between 1 and 2, including dimension 2.

Findings

All arcs in certain mixed labyrinth fractals can have box counting dimension 2.

A family of patterns can produce fractals with any box counting dimension between 1 and 2.

Results are extendable to more general fractal settings.

Abstract

Mixed labyrinth fractals are dendrites in the unit square introduced by Cristea and Steinsky. They were studied recently by Cristea and Leobacher with respect to the lengths of arcs in the fractals. In this article we first give a construction method for mixed labyrinth fractals with the property that all arcs in the fractal have box counting dimension 2. Subsequently, we show how a certain family of patterns can provide a mixed labyrinth fractal of any box counting dimension between 1 and 2, which also coincides with the box-counting dimension of the arc between any two distinct points of the fractal. Finally, we show how the results can be extended to a more general setting.