#### On the length of arcs in labyrinth fractals

On the length of arcs in labyrinth fractals
Ligia L. Cristea 0 1
Gunther Leobacher 0 1
0 Institut für Mathematik und Wissenschaftliches Rechnen, Karl Franzens Universität Graz , Hein- richstrasse 36, 8010 Graz , Austria
1 L.L. Cristea is supported by the Austrian Science Fund (FWF), Project P27050-N26, and by the Austrian Science Fund (FWF) Project F5508-N26, which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”. Part of this work was written while she was also supported by the Austrian-French cooperation project FWF I1136-N26. G. Leobacher is supported by the Austrian Science Fund (FWF) Project F5508-N26, which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”. Part of this work was written when G. Leobacher worked at the Department of Financial Mathematics and Applied Number Theory at Johannes Kepler University Linz (JKU)
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 and Steinsky in Geom Dedicata 141(1):1-17, 2009; Proc Edinb Math Soc 54(2):329-344, 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.
B Ligia L. Cristea
Mathematics Subject Classification 28A80 · 05C38 · 28A75 · 51M25 · 52A38
1 Introduction
Labyrinth fractals are fractal dendrites in the plane, that can also be viewed as a special
family of Sierpin´ski carpets. Such carpets are not only studied by mathematicians,
but also by physicists, e.g., as mathematical models for porous materials, rocks, or
disordered media [1,14]. The mathematical objects called labyrinth fractals were
introduced and studied by Cristea and Steinsky [2–4], on the one hand, and on the
other hand in recent research in physics [6,10–12] objects called fractal labyrinths,
strongly related to the labyrinth fractals, are used, as well as the labyrinth fractals
mentioned above, including in [6] the notation and mathematical frame introduced by
Cristea and Steinsky [2,3]. These fractal labyrinths and labyrinth fractals appear in
physics in several different contexts. To the best of our knowledge, they first occured
in the study of anomalous diffusion, and particle dynamics [10]. In [6] they were
used as a tool for processing and analysing planar nanostructures, while in [11] the
authors applied them in the context of fractal reconstruction of complex images, signals
and radar backgrounds. In the very recent article [12] it is shown how the benefits
of the wide simulation abilities of labyrinth fractals were used in oder to create a
software that generates the shape of ultra-wide band fractal antennas, based on the
geometry of labyrinth fractals, as introduced in [2]. Fractal antennas are already known
to have applications, among others, in medicine, and cellular communications on
base stations and mobile terminals, they have been of interest to scientists from the
fields of physics and electronics for the last decade and still are a subject of ongoing
research.
In nature or in technics, objects that can be described by prefractals of labyrinth
fractals occur in various situations: the system of blood or lymphatic vessels in the body
of humans or animals, the leaf veins of plants, river systems, dendrites in the brain,
the electrical discharches (e.g., lightening) on the one hand, and, on the other hand,
systems of irrigation in agriculture, systems of ressources or information distribution,
communication or transport networks. In the context of physics, a fractal labyrinth is
defined [10] as “a connected topological structure with fractal dimension greater than
1 and with the scaling nature of the conducting channels”. Thus the labyrinth fractals
defined by Cristea and Steinsky [2–4] provide a broad class of fractal labyrinths as
described and used in physics and other applied sciences, and which through their
transparent construction method are amenable to rigorous mathematical treatment. The
results found for these mathematical objects have both potential and actual applications
in, and implications to, fields where their finite,“real” counterparts occur, like, e.g.,
physics, material science, or life science.
Mixed labyrinth fractals were introduced and studied in more recent work by Cristea
and Steinsky [4]. They are a generalisation of the self-similar labyrinth fractals
introduced and studied by the same authors in previous work [2,3]. In the case of mixed
labyrinth fractals more than one pattern is used in order to construct the set, as described
in Sect. 2. It has been proven [4] that, when passing from the self-similar case to the
generalised case of the mixed labyrinth sets and mixed labyrinth fractals, several of
the topological properties are preserved: the mixed labyrinth fractals are dendrites in
the unit square, too, that have exactly one exit on each side of the unit square. In the
self-similar case it was shown that special patterns, called blocked patterns, generate
fractals that are dendrites with the property that the arc between any two points in the
fractal has infinite length.
In the present article we show that in the case of mixed labyrinth fractals the situation
is much more complex: on the one hand, one can find sequences of blocked labyrinth
patterns that generate labyrinth fractals where the arc between any two points in the
fractal is finite, and on the other hand one can find sequences of blocked labyrinth
patterns whose resulting labyrinth fractal has the property that the arc between any
two points of the fractal has infinite length. Moreover, we give an example for the
construction of mixed labyrinth fractals where some arcs in the fractal have finite
length and others have infinite length, analogous to the case when self-similar labyrinth
fractals are generated by a pattern that is horizontally but not vertically, or vertically
but not horizontally blocked (see, e.g., [2]). Finally, we state a conjecture on lengths
of arcs in mixed labyrinth fractals, for future research.
The results in this article provide ideas and modalities for constructing such fractal
dendrites with desired properties regarding the lengths of arcs beween points in the
fractal, that could serve as models, e.g., in the context of particle transport,
nanostructures, image processing. We remark here that although there are several well known
examples of continuous curves with infinite length, like the Peano curve [9], the Hilbert
[7] or the von Koch curve [16, 17], not all of them have the property that the arc between
any two points of the curve has infinite length, as in the case of the arcs in some of
the labyrinth fractals. Moreover, we note that random Koch curves, i.e., objects that
are related, e.g., to arcs between certain points (exit points) in labyrinth fractals, are
studied with respect to random walks by theoretical physicists in connection with
diffusion processes, e.g., [13]. In this context we also mention diffusion processes of
water in biological tissues. There are many more available examples that support the
idea that labyrinth fractals, whether mixed or self-similar, are mathematical objects
worth understanding with respect to their topological and geometrical properties, with
benefits both in mathematics and in other sciences.
2 Labyrinth fractals
One way to construct labyrinth fractals is with the help of labyrinth patterns. Let
x , y, q ∈ [0, 1] such that Q = [x , x + q] × [y, y + q] ⊆ [0, 1] × [0, 1]. For any point
(zx , z y ) ∈ [0, 1] × [0, 1] we define the function PQ (zx , z y ) = (q zx + x , q z y + y).
For any integer m ≥ 1 let Si, j,m = {(x , y) | mi ≤ x ≤ i+m1 and mj ≤ y ≤ jm+1 } and
Sm = {Si, j,m | 0 ≤ i ≤ m − 1 and 0 ≤ j ≤ m − 1}.
Any nonempty A ⊆ Sm is called an m-pattern and m its width. Let {Ak }k∞=1 be a
sequence of non-empty patterns and {mk }k∞=1 be the corresponding width-sequence,
i.e., for all k ≥ 1 we have Ak ⊆ Smk . We let m(n) = kn=1 mk , for all n ≥ 1. Let
W1 = A1, we call W1 the set of white squares of level 1, and define B1 = Sm1 \W1
Wn =
{ PWn−1 (W )}.
W ∈An,Wn−1∈Wn−1
We remark that Wn ⊂ Sm(n), and we define the set of black squares of level n by Bn =
Sm(n)\Wn. For n ≥ 1, we define Ln = W ∈Wn W . Thus, {Ln}n∞=1 is a monotonically
decreasing sequence of compact sets, and L∞ = n∞=1 Ln is the limit set defined by
the sequence of patterns {Ak }k∞=1.
Figures 1, 2, and 3 show examples of labyrinth patterns and illustrate the first three
steps of the construction of a mixed labyrinth set.
We define, for A ⊆ Sm , G(A) ≡ (V(G(A)), E (G(A))) to be the graph of A, i.e.,
the graph whose vertices V(G(A)) are the white squares in A, i.e., V(G(A)) = A and
whose edges E G
( (A)) are the unordered pairs of white squares, that have a common
side. The top row in A is the set of all white squares in {Si,m−1,m | 0 ≤ i ≤ mn − 1}.
The bottom row, left column, and right column in A are defined analogously. A top
exit in A is a white square in the top row, such that there is a white square in the same
column in the bottom row. A bottom exit in A is defined analogously. A left exit in
A is a white square in the left column, such that there is a white square in the same
row in the right column. A right exit in A is defined analogously. One can of course
define the above notions in the special case A = Wn. In this case the top row (in Wn)
Fig. 1 Three labyrinth patterns, A1 (a 4-pattern), A2 (a 5-pattern), and A3 (a 4-pattern)
Fig. 2 The set W2, constructed
based on the above patterns A1
and A2, that can also be viewed
as a 20-pattern
Fig. 3 A prefractal of the mixed
labyrinth fractal defined by a
sequence {Ak } where the first
three patterns are A1, A2, A3,
respectively, shown in Fig. 1
is called the top row of level n. The bottom row, left column, and right column of level
n are defined analogously.
A non-empty m-pattern A ⊆ Sm , m ≥ 3 is called a m × m-labyrinth pattern (in
short, labyrinth pattern) if A satisfies Properties 1, 2, and 3.
Property 1 G(A) is a tree.
Property 2 Exactly one top exit in A lies in the top row, exactly one bottom exit lies
in the bottom row, exactly one left exit lies in the left column, and exactly one right
exit lies in the right column.
Property 3 If there is a white square in A at a corner of A, then there is no white
square in A at the diagonally opposite corner of A.
Let {Ak }k∞=1 be a sequence of non-empty patterns, with mk ≥ 3, n ≥ 1 and Wn
the corresponding set of white squares of level n. We call Wn an m(n) × m(n)-mixed
labyrinth set (in short, labyrinth set), if A = Wn satisfies Properties 1, 2, and 3. It
was shown [4] that if all patterns in the sequence {Ak }k∞=1 are labyrinth patterns, then
Wn is a labyrinth set, for any n ≥ 1. The limit set L ∞ defined by a sequence {Ak }k∞=1
of labyrinth patterns is called mixed labyrinth fractal.
One can immediately see that in the special case when all patterns in the sequence
{Ak }k∞=1 are identical, L ∞ is a self-similar labyrinth fractal, as defined in [2, 3].
In the following we introduce some more notation. For n ≥ 1 and W1, W2 ∈
V (G(Wn )) we denote by pn (W1, W2) the path in G(Wn ) that connects W1 and W2. A
path in G(Wn ) is called -path if it leads from the top to the bottom exit of Wn . The
, , , and -paths lead from left to right, top to right, right to bottom, bottom to
left, and left to top exit, respectively.
Within a path in G(Wn ) each white square in the path is denoted according to its
neighbours within the path: if it has a top and a bottom neighbour it is called -square
(with respect to the path), and it is called , , , , and -square if its neighbours
are at left-right, top-right, bottom-right, left-bottom, and left-top, respectively. If the
considered square is an exit, it is supposed to have a neighbour outside the side of the
exit. A bottom exit, e.g., is supposed to have a neighbour below, outside the bottom,
additionally to its neighbour that lies inside the unit square.
For more details on labyrinth sets and mixed labyrinth fractals and for results on
topological properties of mixed labyrinth fractals we refer to the paper [4].
3 Existing results on arcs in mixed labyrinth fractals
In this section we list some of the results obtained for mixed labyrinth fractals [4] that
are useful in the context of this paper. We use the notation introduced in the previous
section.
Lemma 1 (Arc Construction) Let a, b ∈ L , where a = b. For all n ≥ 1, there are
∞
Wn(a), Wn(b) ∈ V (G(Wn)) such that
W ∈ pn(Wn(a),Wn(b)) W is an arc between a and b.
We recall from [2] that the squares Wn(a), Wn(b), n ≥ 1 in the above lemma are
chosen in the following way: let W (a) be the set of all white squares in n∞=1 Wn that
contain a. Let W1(a) be a white square in G(W1) that contains infinitely many white
squares of W (a) as a subset. For n ≥ 2, we define Wn(a) as a white square in G(Wn),
such that Wn(a) ⊆ Wn−1(a), and Wn(a) contains infinitely many squares of W (a) as
a subset. Wn(b), for n ≥ 1, is defined in the analogous manner.
Proposition 1 Let n, k ≥ 1, {W1, . . . , Wk } be a (shortest) path in G(Wn) between
the exits W1 and Wk , K0 = W1 ∩ fr([0, 1] × [0, 1]), Kk = Wk ∩ fr([0, 1] × [0, 1]),
where fr(·) denotes the boundary of a set, and c be a curve in Ln from a point of K0
to a point of Kk . The length of c is at least (k − 1)/(2 · m(n)).
Let Tn ∈ Wn be the top exit of Wn, for n ≥ 1. The top exit of L∞ is n∞=1 Tn.
The other exits of L∞ are defined analogously. We note that Property 2 yields that
(x , 1), (x , 0) ∈ L∞ if and only if (x , 1) is the top exit of L∞ and (x , 0) is the bottom
exit of L∞. For the left and the right exit the analogous statement holds. Let n ≥ 1,
W ∈ Wn, and t be the intersection of L∞ with the top edge of W . Then we call t
the top exit of W . Analogously we define the bottom exit, the left exit and the right
exit of W . We note that the uniqueness of each of these four exits is provided by the
uniqueness of the four exits of a mixed labyrinth fractal and by the fact that each
such set of the form L∞ ∩ W , where W ∈ Wn, is a mixed labyrinth fractal scaled by
the factor m(n). We note that we have now defined exits for three different types of
objects, i.e., for Wn (and Ak ), for L∞, and for squares in Wn.
For the corresponding results, in detail, for self-similar fractals we refer to [3].
4 Blocked labyrinth patterns, blocked labyrinth sets and a recent conjecture
We recall that an m × m-labyrinth pattern A is called horizontally blocked if the row
(of squares) from the left to the right exit contains at least one black square. It is
called vertically blocked if the column (of squares) from the top to the bottom exit
contains at least one black square. Analogously we define for any n ≥ 1 a horizontally
or vertically blocked labyrinth set of level n. As an example, the labyrinth patterns
shown in Figs. 1 and 4 are horizontally and vertically blocked, while those in Fig. 5
are not blocked.
In the self-similar case the following facts were proven [3, Theorem 3.18]:
Fig. 5 Examples of labyrinth patterns, that are neither horizontally nor vertically blocked
Fig. 6 An example: the special
cross pattern A1 with m1 = 11
Theorem 1 Let L∞ be the (self-similar) labyrinth fractal generated by a horizontally
and vertically blocked m × m-labyrinth pattern. Between any two points in L∞ there
is a unique arc a. The length of a is infinite. The set of all points, at which no tangent
to a exists, is dense in a.
For the case of mixed labyrinth fractals, Cristea and Steinsky [4] recently formulated
the following conjecture.
Conjecture 1 Let {Ak }k∞=1 be a sequence of both horizontally and vertically blocked
labyrinth patterns, mk ≥ 4. For any two points in the limit set L∞ the length of the
arc a ⊂ L∞ that connects them is infinite and the set of all points, where no tangent
to a exists, is dense in a.
In this article we solve the arc length problem posed by the above conjecture by
showing that, depending on the choice of the both horizontally and vertically blocked
labyrinth patterns in the sequence {Ak }k∞=1, both situations can occur: the arc between
any points of the fractal has finite length, or the arc between any two points of the
fractal has infinite length.
Example 1 Let {Ak }k∞=1 be a sequence of (both horizontally and vertically) blocked
labyrinth patterns, mk ≥ 11, with mk = 2ak + 1, ak ≥ 5.
We consider a sequence of patterns that are both vertically and horizontally blocked
and have a “cross shape” like A1 in Fig. 6, i.e., the pattern looks like a “cross” centered
in the “central square” of the pattern (here coloured in light grey) and each “arm” of
the cross is “blocked” such that in order to get from the “center” of the cross to any
of the four exits of the pattern we have to go a detour around a black square that lies
in the same row or column as the respective exit and the mentioned “central square”
of the pattern. More precisely, we position the four black squares between the central
square and the exits of Ak in the columns (rows) (ak + 1)/2 and (3ak + 3)/2, if ak is
odd, and in the columns (rows) (ak + 2)/2 and (3ak + 2)/2, if ak is even. We call these
patterns special cross patterns. One can immediately see that the “central square” of
such a special cross pattern, where the four “arms” of the cross meet, changes its type,
depending on the path in G(A1) that we consider between two exits of the pattern A1:
in the -path in the pattern, it is a -square, and in the y-path, it is a y-square, for any
y ∈ { , , , , }.
We recall that the path matrix of a labyrinth set or a labyrinth pattern A is a 6 ×
6matrix M such that the element in row x and column y is the number of y-squares in the
x -path in G(A). It was proven [4, Proposition 1] that, for any sequence of labyrinth
patterns {Ak }k≥1 with corresponding sequence of path matrices {Mk }k≥1, for any
integer n ≥ 1 the matrix M (n) := kn=1 Mk is the matrix of the mixed labyrinth set
Wn (of level n), i.e., the sum of the entries in any row of M (n) gives the length of
the path between two of the exits in G(Wn ). For more details and properties of path
matrices we refer to the papers [2–4].
With the help of Figs. 6 and 7 one can easily check that for this special sequence of
labyrinth patterns {Ak }k≥1, the path matrix of the pattern Ak is
⎛ 2ak − 3 0 2 2 2 2 ⎞
⎜ 0 2ak − 3 2 2 2 2 ⎟
Mk = ⎜⎜⎜⎜ aakk −− 22 aakk −− 22 23 32 22 22 ⎟⎟⎟⎟ , for k ≥ 1 and mk = 2ak + 1.
⎝⎜ aakk −− 22 aakk −− 22 22 22 23 32 ⎠⎟
of the top and of the bottom edge of W . We proceed analogously in the case when W
is a square of type , , , , , in each case γnq |W is the union of two line segments
(both horizontal, or one horizontal and one vertical) that both go through the center
of W and the midpoint of some edge of W , such that the sum of their lengths is m(1n) .
We immediately get the length of the curve γn, for n ≥ 1:
k=1
k=1
4
1 + mk
Now we study the sequence { (γn)}n≥1. From (2) we easily see that { (γn)}n≥1 is
a strictly increasing sequence, thus limn→∞ (γn) = supn=1,2,... (γn).
By basic mathematical analysis facts kn=1(1 + m4k ) converges if and only if
converges, i.e.,
4
k≥1 mk
k≥1 m1k < ∞. By taking, e.g., ak = 5k , for k = 1, 2, . . . we obtain
supn=1,2,... (γn) = limn→∞ (γn) < ∞.
Remark One can verify, by using the definition of the Hausdorff distance dH , that, for
q ∈ E , the arc γ q in L∞ that connects the two exits of L∞ indicated by q satisfies
dH (γnq , γ q ) → 0, for n → ∞. Here we mean the Hausdorff distance between the
images of the two curves, as sets in the Euclidean plane endowed with the Euclidan
distance.
Lemma 2 With the above notation, there are parametrisations γ˜nq (t ) : [0, 1] →
[0, 1]2 and γ˜ q (t ) q: [0, 1] → [0, 1]2 of γnq and γ q , respectively, such that for all
q ∈ E , we have ||γ˜n − γ˜ q ||∞ → 0, for n → ∞, where || · ||∞ is the supremum norm.
the left aTnhdisrpigichttuerxeisthoofwassaqufraargemWen∈toWfγnnq, ,inantdhe(dpaicshtuerde)tγhneq−m1o.Hstelreeftqan∈dEmionsdticraigtehst tdhoatttethdepaoricnctsonnects
Fig. 9
Lemma 3 Let γ˜n : [0, 1] → [0, 1]2 and γ˜ : [0, 1] → [0, 1]2 be parametrisations of
the planar curves γn and γ , respectively, whose lengths we denote by (γn) and (γ ). If
||γ˜n −γ˜ ||∞ → 0, for n → ∞, and supn (γn) < ∞, then (γ ) < ∞. Moreover, in this
case the following inequalities hold: lim infn→∞ (γn) ≤ (γ ) ≤ lim supn→∞ (γn).
of a curve, (γn) ≥
following inequalities:
Proof We give an indirect proof of the first assertion of the lemma, the second
one we leave as an exercise, since for our purposes the first assertion is already
enough. Assume (γ ) = ∞. Since supn (γn) < ∞, we can choose a positive
integer N such that N > sup (γn) + 1. Then, by the definition of the length of
n
a curve, there exist the real numbers 0 = s0 < s1 < · · · < sm = 1, m ≥ 1,
such that km=1 ||γ˜ (sk ) − γ˜ (sk−1)|| > N , where || · || denotes the Euclidean norm
in the plane. From the convergence hypothesis it follows that there exists an
integer n0 such that for every n ≥ n0 we have ||γ˜ − γ˜n||∞ < 2(m1+1) , and thus
1
k=m1,a..x.,m ||γ˜ (sk ) − γ˜n(sk )|| < 2(m + 1) . Moreover, by the definition of the length
m
k=1 ||γ˜n(sk ) − γ˜n(sk−1)||. Thus, we now easily obtain the
k=0
≤ 2
k=1
which leads to a contradiction.
Now, for an arbitrary n ≥ 1, let us take W ∈ Wn, L∞|W := L∞ ∩ W and consider
any two of the exits e1, e2 of the square W (as defined in the paper [4]). Then the arc
in L∞|W that connects e1 and e2 is the scaled image of the arc between two exits (of
the same types) of a labyrinth set L∞ generated by the sequence of patterns {A k }k∞=1,
where A k = Ak+n, and the scaling factor is m(n). Therefore, one can easily see that
the arc beween any such exits of any square W ∈ Wn, for any n ≥ 1, is finite.
Herefrom it then easily follows that if x , y are points that belong to the set of points
in L∞ that consists of all centres and all exits of squares of ∪n≥1V(G(Wn)), then the
length of the arc in L∞ that connects x and y is finite.
Let En be the set of all points of L∞ that are exits of squares of level n, and Cn
be the set of all points of L∞ that are centers of squares of level n. For any two
distinct points x , y ∈ L∞, we introduce the notation a(x , y ) for the arc in L∞ that
connects the points x and y . Let now W ∈ V(G(Wn)), with n ≥ 0 (for n = 0, W
is the unit square, otherwise it is a white square of level n, as defined before). Let c
be the center of W, and e one of its four exits. Now, we want to show that for any
point x ∈ (IntW ∩ L∞)\ ∪k≥n (Ek ∪ Ck ), where Int denotes the interior of the set,
(a(x , c)) < ∞ and (a(x , e)) < ∞. In the following we give a proof of the first
inequality. Therefore, we consider two cases.
First, we assume that x is a point on one of the four “main arms” of L∞ ∩ W (which
is in fact the scaled image of the mixed labyrinth fractal defined by the sequence
{Ak }k≥n+1), i.e., x lies on the arc in L∞ that connects the center of W with one of its
exits. In this case, it easily follows from the above results that the length of the arc in
L∞ between x and the center of W has finite length (that is less than one half of the
length of the arc between two exits in W ).
In the second case, we assume that x does not lie on a “main arm” of L∞ ∩ W ,
i.e., x lies on a “branch” of the dendrite, that originates at a point, say c , with c ∈
k≥n+1 Ck , that lies on one of the four “main arms” of L∞ ∩ W (that connects the
center c ∈ Cn of W with one of its exits, say e ∈ En). By the construction of the
fractal and of arcs in the fractal (Lemma 1), there exists a point e ∈ k≥n+1 Ek
such that x lies on the arc a(c , e ) in L∞ that connects c and e , which, due to the
above considerations, has finite length. Since (a(c, x )) = (a(c, c )) + (a(c , x )) ≤
(a(c, c )) + (a(c , e )) = (a(c, e )) < ∞, it follows that (a(c, x )) < ∞. We leave
the proof of the inequality (a(x , e)) < ∞ to the reader as an exercise.
Let now x , y ∈ L∞ be two distinct points, and let Wn(x ) and Wn(y) be two squares
in Wn such that x ∈ Wn(x ) and y ∈ Wn(y). The squares Wn(x ), n ≥ 1 are chosen in
the following way: let W (x ) be the set of all white squares in n∞=1 Wn that contain
x . Let now W1(x ) be a white square in G(W1) that contains infinitely many white
squares of W (x ) as a subset. Now we define, for n ≥ 2, Wn(x ) as a white square in
G(Wn), such that Wn(x ) ⊆ Wn−1(x ), and Wn(x ) contains infinitely many squares of
W (x ) as a subset. We define Wn(y), for n ≥ 1, in the analogous manner. Let pn be the
path between Wn(x ) and Wn(y) constructed as in Lemma 1. Since x = y, it follows
that there exists an integer n ≥ 1, such that pn consists of at least 3 squares. Let then
W ∈ pn be a square with W ∈/ {Wn(x ), Wn (y)}. By the construction of the arc a in
L∞ between x and y as described in Lemma 1, a ∩ W is an arc between two exits
of W, and thus has finite length. Since a(x , y) is the union of finitely many such arcs
of finite length with the arcs a(x , ex ) and a(ey , y), where ex ist one of the exits of
Wn(x ) and ey is one of the exits of Wn(y), namely {ex } = a(x , y) ∩ fr(Wn(x )), and
{ey } = a(x , y) ∩ fr(Wn(y)), it follows that a(x , y) has infinite length.
The above example shows that one can find a sequence of patterns that generates a
mixed labyrinth fractal with the property that the length of the arc that connects any
two points in the fractal is finite. Moreover, one can see that for a labyrinth pattern
that contains such a “special cross” the length of the paths between the exits of the
pattern does not change, it is the same as here, and thus the arc lengths in the fractal
also remain finite, as in the above example.
Thus we have proven the following result.
Proposition 3 There exist sequences {Ak }k∞=1 of (both horizontally and vertically)
blocked labyrinth patterns, such that the limit set L∞ has the property that for any
two points in L∞ the length of the arc a ⊂ L∞ that connects them is finite.
Based on a theorem in the book of Tricot [15, p.73, Chap. 7.1] regarding the existence
of the tangent to a curve of finite length, we obtain the following stronger result:
Theorem 2 There exist sequences {Ak }k∞=1 of (both horizontally and vertically)
blocked labyrinth patterns, such that the limit set L∞ has the property that for any
two points in L∞ the length of the arc a ⊂ L∞ that connects them is finite. For almost
all points x0 ∈ a (with respect to the length) there exists the tangent at x0 to the arc a.
1. It is easy to see that such special cross patterns as shown in Fig. 6, with such
a “detour” on each of the four arms, exist only for width m ≥ 11. Moreover,
one can easily check that for the above example both the box-counting and the
Hausdorff dimension of the fractal is dimB (L∞) = dim H (L∞) = 1 and also the
box-counting dimension of any arc that connects a pair of exits in L∞ is 1. The
same holds for the arc between any two distinct points in the fractal.
2. By the definition of a mixed labyrinth fractal, by the shape of special cross patterns,
and the arc construction given in Lemma 1, one can check that the fractal is the
countable union of rectifiable 1-sets. An example of such a countable collection
of rectifiable 1-sets is as follows: take, for any level n ≥ 1 of the construction,
the arcs in L∞ that connect the center of any square W ∈ G(Wn) and any of the
midpoints of its sides, i.e., any of the four exits of W, as well as the four arcs in
L∞ that connect the center of the unit square with any of its midpoints (the four
exits of the mixed labyrinth fractal).
3. In Example 1 we could also take, e.g., cross patterns with ak = 2k , for k ≥ 1, and
consider the first two patterns in the sequence of generating patterns, to be just
unblocked, symmetric cross patterns, with the width mk = 2ak + 1, k ∈ {1, 2},
and for k ≥ 3 special cross patterns. Then, Wn is blocked for all n ≥ 3, and the
resulting limit set L∞ would still have the property that the arc between any two
points in the fractal has finite length.
In the following example we show that one can use special cross patterns like the one
shown in Fig. 6 in order to construct mixed labyrinth fractals with the property that
the arc between any two points in the fractal has infinite length.
Example 2 Let {Ak }k∞=1 be a sequence of special cross patterns like those occurring in
Example 1, r ≥ 2 be an arbitrarily fixed integer, and let {Ai , i = 1, . . . , r } ⊂ {Ak }k≥3
be a finite set of blocked labyrinth patterns among those in the above infinite sequence,
where mi = 2ai + 1 denotes the width of the pattern Ai , and li = 2ai + 5 is the length
of the path between any two exits in G(Ai ), for i = 1, . . . , r . We define a new sequence
of labyrinth patterns {A∗j } j≥1, e.g., in the following way: A∗j ∈ {Ai , i = 1, . . . , r }.
Let L∞ be the mixed labyrinth fractal generated by the sequence of patterns {A∗j } j≥1,
and let a∗ := max{ai , i = 1, . . . , r }, m∗ := 2a∗ + 1, and l∗ := 2a∗ + 5. Since
l∗
m∗
4
1 + m∗
→ ∞, for n → ∞,
Fig. 10 An example: a
half-blocked cross pattern A1
with width m1 = 11 that is
horizontally blocked, but not
vertically blocked
it follows from Lemma 1 that the length of the arc between any two exits in L ∞ is
infinite. By using arguments analogous to those in Example 1, one can show that the
infinite length of the arc between any two exits of L ∞ implies that the arc between
any two points in the fractal is infinite, as in the case of self-similar labyrinth fractals
generated by both horizontally and vertically blocked patterns [2, 3].
Thus we have proven the following
Proposition 4 There exist sequences {Ak }k∞=1 of (both horizontally and vertically)
blocked labyrinth patterns, such that the limit set L ∞ has the property that for any
two points in L ∞ the length of the arc a ⊂ L ∞ that connects them is infinite.
Example 3 In Fig. 10 we have a “half-blocked” labyrinth pattern with width 11,
that is horizontally, but not vertically blocked, we call such a pattern (of width
m = 2a + 1 ≥ 11) a half-blocked cross pattern, where either the horizontal or
the vertical arms of the cross make a detour around a black square, positioned
as in the case of the special cross patterns used in Example 1, and the “central
square” of the pattern (where all “cross arms” meet) is positioned in column and
row a + 1. Suppose we have a sequence {Ak }k∞=1 of such patterns, with mk = 2ak + 1,
and ak ≥ 5, for k ≥ 1. For any pattern Ak of the above sequence, the path
matrix is
⎛ 2ak + 1
0
ak
ak
ak
ak
Thus, the lengths of the paths between exits in G(Ak ) are: k = 2ak +1, k = 2ak +5,
k = k = k = k = 2ak + 3. One can immediately check that the length of the arc
between the top and bottom exit of the resulting labyrinth set L ∞ is 1, no matter how
we chose the sequence {ak }k≥1. Now, let us analyse the arc between the left and the
rfiogrhkt e≥xi1t,inthLen∞: bk∞e=tw1eme1kn =the∞le.ft and right exit in Wn . If we choose, e.g., ak = k + 4,
From Proposition 1 we have, with the notation used above: for q ∈ {
1 n
= 2
k=1
4
1 + mk
− 2
1
n
k=1 mk
Under the above assumptions, limn→∞( 21 kn=1 1 + m4k − 2 kn1=1 mk ) = ∞, for
n → ∞, as one can immediately check. Herefrom one can easily infere that also the
arcs in L∞ that connect the top or the bottom exit of L∞ with the left or the right exit
of L∞ have, all four, infinite length.
Moreover, one can show, by using arguments analogous to those mentioned when
proving Proposition 3, that for any W ∈ Wn the arc L∞ that connects the left and the
right exit of W has infinite length. This also holds for the arc in L∞ that connects the
top or the bottom exit of W with the left or the right exit of W .
Proposition 5 There exist sequences of horizontally blocked and not vertically
blocked labyrinth patterns, such that the resulting mixed labyrinth fractal L∞ has
the following properties:
1. The arc in the fractal that connects the top and the bottom exit of L∞ has finite
length (equal to 1). The arc in the fractal that connects the top and bottom exit of
any square in Wn is a vertical segment with finite length. Any vertical line segment
that is contained in the fractal has finite length.
2. The arc in the fractal that connects the left and the right exit of L∞ has infinite
length. The arc in the fractal that connects the left and right exit of any square in
Wn has infinite length.
3. The arc in the fractal that connects the exit e1 and the exit e2 in L∞, where e1 is
either the top or the bottom exit, and e2 either the left or the right exit, has infinite
length, and the same holds for the arcs between these pairs of exits in any square
in Wn.
The corresponding analogous statements regarding the existence of sequences of
vertically and not horizontally blocked labyrinth patterns hold, such that L∞ has the
corresponding analogous properties.
For the above sequence of half-blocked cross patterns the length of the arc in L∞
that connects two arbitrarily chosen distinct points x , y ∈ L∞ has finite length if and
only if it is contained in a vertical line segment which is, itself, a subset of L∞.
Some final remarks In the case of mixed, i.e., not self-similar labyrinth fractals, not
just the shape of the patterns, but also their width plays an essential role as a parameter
that influences the lengths of arcs between exits or between any points in the fractal.
Conjecture A sequence of both horizontally and vertically blocked labyrinth patterns
with the property that the sequence of widths {mk }k≥1 is bounded, generates a mixed
labyrinth fractal with the property that for any x , y ∈ L∞ the length of the arc in the
fractal that connects x and y is infinite.
Acknowledgements Open access funding provided by Austrian Science Fund (FWF). The authors thank
Bertran Steinsky for valuable remarks on the manuscript. We thank the referee for helpful comments.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
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