Mixed labyrinth fractals

Abstract

Labyrinth fractals are self-similar fractals that were introduced and studied in recent work [2, 3]. 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 Sierpiński carpets.

Triangular Labyrinth Fractals

“…The proof of the following two propositions works analogously to the case of mixed labyrinth fractals [5]. Proposition 5.…”

“…Proof. The proof, which uses the Hahn-Mazurkiewicz-Sierpiński theorem [12, Theorem 2, p.256] as well as the Jordan Curve theorem, is almost identical to the the proof of Theorem 1 in [5] but uses Corollary 1 in place of the corresponding result there.…”

“…Moreover, the arc a(l ∞ , b ∞ ) in L ∞ is a union of arcs between exits in some of the mentioned copies of L ∞ . Since one of the squares in the path of type in G(W 1 ) is of type , it follows by [5,Proposition 3] or by Proposition 5 applied to mixed labyrinth fractals, that the arc a(l ∞ , b ∞ ) in L ∞ has as subarc a copy (scaled by factor 1/m 1 ) of the arc between the left and bottom exits in L ∞ , which has infinite length. Therefore, it follows that (a(l ∞ , b ∞ )) = ∞.…”

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Labyrinth fractals are dendrites in the unit square. They were introduced and studied in the last decade first in the self-similar case [3,4], then in the mixed case [5,6]. Supermixed fractals constitute a significant generalisation of mixed labyrinth fractals: each step of * L.L.

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On certain families of planar patterns and fractals

“…In more recent work [14] dedicated to mixed labyrinth fractals, Steinsky and Cristea gave an other sufficient condition for the total disconnectedness of certain classes of generalised Sierpiński carpets, that occur in relation labyrinth fractals that we present in Section 5. We note the combinatorics-flavoured approach of carpets by identifying special families among the generating patterns that provide certain properties, which was inspired by the results that were obtained for limit net sets, that is different from the approaches of other authors who have studied similar objects.…”