Curves of infinite length in 4 × 4-labyrinth fractals

Abstract

We study 4 × 4-labyrinth fractals, which are self similar dendrites. For all 4 × 4-labyrinth fractals we answer the question, whether there is a curve of finite length in the fractal from one point to another point in the fractal. In the first case, between any two points in the fractal there is a unique arc a, the length of a is infinite, and the set of points, where no tangent exists to a, is dense in a. In the second case, there are also pairs of points between that there is a unique arc of finite length.

Curves of infinite length in labyrinth fractals

“…All the results in this section have been proven by Cristea and Steinsky [5]. Nevertheless, we present them here, as they will be used later.…”

“…In this section it will be helpful to keep in mind the example in Figure 1, which we have already discussed [5].…”

“…We note that the graph concepts we use here were defined by Cristea and Steinsky [5]. For n 1, we define G(W n ) ≡ (V(G(W n )), E(G(W n ))) to be the graph of W n , i.e.…”

“…Having discussed 4 × 4-labyrinth fractals in [5], we now devote our attention to m × mlabyrinth fractals for m 5. These fractal sets are constructed iteratively, as described in § 1.1.1.…”

On the length of arcs in labyrinth fractals

“…the sum of the entries in any row of M (n) gives the length of the path between two of the exits in G(W n ). For more details and properties of path matrices we refer to the papers [2,3,4].…”