# Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

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## Abstract

**:**

## 1. Introduction

**Elementary fractal geometry.**Dimension and measure play a dominant part in fractal geometry. Most papers consider general classes of fractals in a measure-theoretical setting and prove results which hold for almost all representatives of this class [1,2]. However, there are also specific fractals with special symmetries and properties related to polynomial equations, algebraic numbers, and certain graphs and automata. We call the study of such examples elementary fractal geometry. This is similar to classical geometry where we can study arbitrary domains in ${\mathbb{R}}^{d}$ or regular polyhedra, say. The pentagon, associated with $\sqrt{5},$ has led the way to Penrose tiles, and to the fractals in this paper.

**Self-similar sets and open set condition.**A self-similar set is a nonempty compact subset A of ${\mathbb{R}}^{d}$ which is the union of shrinked copies of itself. This is expressed by Hutchinson’s equation

**The aim of this paper.**Sierpiński constructed his triangle and carpet more than 100 years ago as topological counterexamples [11]. After 1980, physicists took them as models of porous materials, and mathematicians developed an analysis of the heat equation, Brownian motion, and eigenvalues of the Laplace operator on just these spaces. See the books by Kigami and Strichartz [12,13] for an introduction, and the literature cited there. However, both spaces have properties which rarely occur in nature:

- They have a few characteristic directions.
- They contain line segments.
- Their holes have a small perimeter compared to their area.

- Have no characteristic directions and an isotropic type of symmetry;
- Contain no line segments, and
- have holes with large perimeter and rather complicated shape.

**Algebraic OSC.**Let ${F}^{\ast}$ denote the semigroup of similitudes generated by the IFS $F.$ There is an algebraic equivalent for OSC, formulated explicitly in terms of F [1,9].

**Neighbor complexity.**As a rigorous concept of complexity, we take the number of neighbor maps. Consider the map $h={f}_{w}^{-1}{f}_{v}$ where $w,v\in {K}^{n}$ denotes different words of length n from the alphabet $K.$ It is an isometry. We call h a neighbor map or neighbor type if the corresponding pieces of A intersect: ${f}_{w}\left(A\right)\cap {f}_{v}\left(A\right)\ne \varnothing .$ The number of neighbor types will be taken as the complexity of the IFS $F.$ If the number is finite, we say that F is of finite type. As a consequence, OSC is fulfilled: a special open set was constructed in [20].

**Approaches to non-crystallographic patterns.**There are various ways to generalize crystallographic finite type systems. One way is projection of integer data from a higher-dimensional space, known from quasiperiodic tilings used for modelling quasicrystals [22,23]. This is implemented in IFStile. Another possibility is to replace the Equation (1) with a system of equations for different types of sets, which is called a graph-directed construction [24]. In their study of self-similar tilings, Thurston, Kenyon, and Solomyak [25,26,27]. studied graph-directed systems without any symmetries. They assume that there are finitely many tiles up to translation. This approach includes the case of rotations by rational angles—rational multiples of ${360}^{\circ}$. A self-similar set A with pieces rotated by multiples of ${60}^{\circ}$, as in the middle of Figure 1, can be generated by a translationally finite graph-directed system of six sets without considering rotations.

**Contents of the paper.**In the translationally finite case as well as in self-similar sets with crystallographic data, including those which are projected from higher-dimensional lattices, all motives appear in a finite number of directions. Here we construct fractals with a dense set of characteristic directions, in other words with no characteristic directions at all. The bottom of Figure 1 and Figure 2 give a first impression of our patterns. We replace integer data with rational data, where no theorems guarantee the existence of finite type examples, except for trivial cases such as Cantor sets. We performed an extensive computer search, checking some hundreds of millions of IFS, and present selected results.

**Motivation.**Beside the potential of isotropic fractals for modelling in science, there are various mathematical reasons for this research. One is pure curiosity: to see what is beyond crystallographic symmetries. Another motivation is to show that our approach with symmetries is much wider than the translationally finite setting. Moreover, fractals without characteristic directions show some measure-theoretic uniformity. They have “statistical circular symmetry”, as certain quasiperiodic tilings, and physical materials of this type show a diffraction spectrum of rings [28,30]. Their projections onto lines possess the same dimension for every direction, while in general we have an exceptional set of directions with a smaller dimension. Further uniformity properties were proved mainly by Shmerkin and coauthors, see [31,32,33,34]. Here we construct concrete examples of such sets which also have attractive topological properties.

## 2. Basic Assumptions and a Simple Example

**Basic assumptions.**We recall the basic Equations (1) and (4):

**An example with irrational rotation.**Before we discuss the difficulties with the base $B,$ we consider examples where the standard base can be taken as $B.$ Let

## 3. The Neighbor Graph

**Definition.**We shall now determine the combinatorial structure of our example IFS, the so-called neighbor graph. For the case of pieces of equal size, this object has been defined in various papers, including [37,38,39,40,41,42,43,44,45,46,47]. We introduce the concept briefly and refer to the literature for details. The IFStile package determines neighbor graphs also for IFS with different contraction ratios and for graph-directed constructions.

**Intuitive construction.**The first level intersections all involve ${A}_{3}$. Initial edges labelled 1,3; 3,1; 3,2 and 3,4 go to the following vertices, respectively.

**The computer algorithm.**A computer easily generates lots of neighbor maps ${h}_{wv}={f}_{w}^{-1}{f}_{v}$ by repeatedly applying the recursive formula ${h}^{\prime}={f}_{k}^{-1}h{f}_{j}$ with $k,j\in \{1,\phantom{\rule{0.166667em}{0ex}}\dots ,m\}.$ The problem is to decide which of these actually fulfil ${f}_{w}\left(A\right)\cap {f}_{v}\left(A\right)\ne \varnothing .$

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

## 4. What We Can Conclude from the Neighbor Graph

**Topology-generating automaton.**The neighbor graph is an automaton which generates the topology of $A.$ All topological properties of A are encoded in the neighbor graph. In the argument above, we pretended that we can see that the pieces ${A}_{1}$ and ${A}_{3}$ intersect. This was not true. In fact, pictures did repeatedly lead us to wrong conclusions. Only the calculation of the neighbor graph can verify that ${A}_{1}$ and ${A}_{3}$ have common points.

**Connectedness properties.**Recall that the neighbor graph G is constructed so that there is an outgoing edge from each vertex, and hence an infinite directed path starting in each vertex. Thus, there is an initial edge with label $k,j$ if and only if ${A}_{k}\cap {A}_{j}\ne \varnothing .$ So, from G we can define the connectedness graph ${G}_{c}$ of $A,$ with vertex set $\{1,\phantom{\rule{0.166667em}{0ex}}\dots ,m\}$ and undirected edges between $k,j$ whenever ${A}_{k}$ intersects ${A}_{j}.$ For our example, the connectedness graph is a three-star, or letter Y, with central vertex three. The following is known. The first assertion is a classical theorem of Hata, proved by inductively constructing chains of intersecting pieces.

**Proposition**

**3.**

- (i)
- The attractor A is connected if and only if ${G}_{c}$ is connected.
- (ii)
- Suppose A is connected. Then A contains a closed Jordan curve if only if either ${G}_{c}$ is connected, or two pieces ${A}_{j},{A}_{k}$ intersect in more than one point.
- (iii)
- Let $h={f}_{k}^{-1}{f}_{j}$ be the vertex of the neighbor graph G which corresponds to $D={A}_{k}\cap {A}_{j}\ne \varnothing .$ Let $\mathcal{C}$ denote the set of all directed cycles which can be reached by a path from vertex $h.$ The intersection D is a singleton if and only if $\mathcal{C}$ consists of one element ${C}_{1}$ and there is only one path from h to ${C}_{1}.$ The intersection is finite if there is no path between ${C}_{1}$ and ${C}_{2}$ for any ${C}_{1},{C}_{2}\in \mathcal{C}.$ The set D is uncountable if there exist ${C}_{1},{C}_{2}\in \mathcal{C}$ and paths from ${C}_{1}$ to ${C}_{2}$ and back. Otherwise, D is countably infinite.

**Cutpoints and first-level intersections.**Consider the fixed point $y=0$ of ${f}_{3}$ in the above example. It does not belong to two pieces since the address $\overline{3}=333\phantom{\rule{0.166667em}{0ex}}\dots $—even the word 33—is not labelling a path in the neighbor graph. Nevertheless, y is an important point for the topology of $A.$ It is a cutpoint of degree 3: $A\backslash \left\{y\right\}$ consists of three connected components. This can be easily seen: removal of ${A}_{3}$ from A results in three different pieces, and removal of ${A}_{33}$ from A results in three larger pieces, and so on. Thus A involves a rather simple tree structure, despite containing Cantor set intersections and closed Jordan curves. We call such a space a web.

**Hausdorff dimension of the boundary.**Apart from connectedness properties, let us remind the fact that the neighbor graph verifies that our IFS is of finite type and fulfils the OSC. In particular, the Hausdorff dimension of A equals [1,2,7]

## 5. Algebraic Aspects of the Neighbor Graph

**An open question.**The neighbor graph completely describes the topology of $A,$ but not the Hausdorff dimension of $A.$ It is well-known that the dimension of a Koch curve can be changed without changing the topology, just by varying the angle of the maps [7]. So the question is which properties should be added to the neighbor graph in order to completely characterize the IFS, up to isomorphy. Isomorphy involves change of the coordinate system and permutation of the maps, see [4]. The question seems difficult, so we discuss it only for our simple example.

**Generating relations.**The neighbor maps generate a group of isometries. Whenever we extend the self-similar construction of a connected attractor A to the outside, by forming supertiles ${f}_{{k}_{1}}^{-1}\left(A\right),\phantom{\rule{4pt}{0ex}}{f}_{{k}_{2}}^{-1}{f}_{{k}_{1}}^{-1}\left(A\right),\dots ,$ any isometry between two ‘tiles’ of such a pattern will belong to that group. In algebra, groups are often defined by a system of generators and generating relations. It turns out that the neighbor graph provides such relations between the neighbor maps. In this way it gives algebraic information on the IFS beyond the topology of $A.$ Usually the equations are highly nonlinear. For our simple example we can separate and solve them, however.

**Finding IFS data from a neighbor graph.**Let us assume that we have an IFS given by $g,{h}_{1},\dots ,{h}_{4}$ which produces the neighbor graph of Figure 4. For simplicity we assume ${h}_{3}=id$, which can be arranged by passing to the isomorphic IFS $\tilde{g}={h}_{3}^{-1}g,\phantom{\rule{4pt}{0ex}}\tilde{{h}_{j}}={h}_{3}^{-1}{h}_{j}.$ Then the initial edges 3,1, 3,2 and 3,4 imply that ${h}_{1}={n}_{1},\phantom{\rule{4pt}{0ex}}{h}_{2}={n}_{2},$ and ${h}_{4}={n}_{4}.$

**Proposition**

**4.**

- (i)
- ${h}_{2}\left(z\right)={w}_{2}-z$ and ${h}_{4}\left(z\right)={w}_{4}-z$ where ${w}_{2},{w}_{4}\in \mathbb{C}$ fulfil ${w}_{4}=(2-\lambda ){w}_{2}.$
- (ii)
- $v=t{w}_{2}.$
- (iii)
- $t(4-\lambda )=\lambda ({\lambda}^{2}-3\lambda +3)\phantom{\rule{4pt}{0ex}}.$

**Proof.**

**Proposition**

**5.**

## 6. Examples from Pythagorean Triples

**Pythagorean triples and irrational rotation.**A rotation of the plane by an angle $\alpha $ is called rational if $\alpha =\frac{k}{n}\xb7\pi $ for integers $k,n,$ and irrational otherwise.

**Proposition**

**6.**

**Proof.**

**The challenge.**We briefly explain the goal of our experiments. For crystallographic data, each integer expansion matrix M leads to IFS with OSC and $m=detM.$ This is proved by taking so-called complete residue systems as ${h}_{1},\dots ,{h}_{m}$ [19], and it is well-known that the attractors are tiles. By dropping one or more of the mappings we can then easily create carpets, as in Figure 1. For irrational data, however, only very few tiles are known [5,28,29,30], and we found no further tiles in our experiments.

**Results of the computer experiments.**While a computer search with rational rotations gives thousands of examples within a minute, many of them with high complexity [4], the interactive search for examples with irrational IFS is more difficult. First we obtain only Cantor sets without any intersections. Then few examples may have pieces with intersections, creating a non-trivial neighbor graph, but A will still remain a Cantor set. Then, by modifying the best datasets, we obtain some fractals which are connected, or have at least connected subsets. Finally, deleting all bad examples and modifying only the best ones, we obtain carpets with uncountable intersections in most of our experiments. As a rule, they have low complexity, between 10 and 40 neighbor types. The number of such carpets varied between 1 and 100, for different g and $s.$

## 7. An Example from the Hexagonal Lattice

**Proposition**

**7.**

**Proof.**

## 8. Classification of Fractal Patterns

**Principles.**Now we shall try to insert some order into our zoo of fractal examples. As in biology and crystallography, we have to divide them into species and families. This will be conducted in three steps.

- We fix the lattice, which corresponds to the species. We can choose the square lattice with the basic ${90}^{\circ}$ rotation, or the hexagonal lattice with its ${60}^{\circ}$ rotation. Other choices are given below. Algebraically, the lattice is induced by a number field which specifies algebraic numbers such as i or $\sqrt{3}$ which we need beside rational numbers. This number field comes with a symmetry group consisting of rotations defined by the number of modulus one. There can be an number of infinite rotations with infinite order.
- We choose an expansion map $g\left(x\right)=Mx$ or $g\left(z\right)=\lambda z.$ The number $\lambda $ is taken as an algebraic integer of norm greater than one in our number field, sometimes as an algebraic rational. Then, $\sqrt{detM}$ or $\left|\lambda \right|$ is the contraction factor r of the IFS. Since we require OSC, the number of maps in the plane is bounded by $m{r}^{2}\le 1,$ or $m\le detM.$
- The choice of lattice and expansion, together with a finite selection of rotations $s,$ determines the family of fractals. In the last step, concrete instances of this family are produced by choosing particular ${s}_{k},{v}_{k}$ for $k=1,\dots ,m$ such that the OSC is fulfilled.

**Examples.**In Figure 1, the Sierpiński triangle was drawn in a symmetric way. When we make use of this symmetry, or include ${120}^{\circ}$ rotations in our IFS, we are in the hexagonal lattice, or the number fields $\mathbb{Q}\left(\sqrt{-3}\right),$ and have a symmetry group of three rotations. However, usually we generate a symmetric fractal in its most simple form. That is, ${f}_{k}\left(x\right)=(x+{c}_{k})/2$ where ${c}_{k}$ is the fixed point of ${f}_{k}.$ Then the symmetry group is trivial, ${s}_{k}=id$ for all $k,$ and all choices of non-collinear points ${c}_{k}$ are isomorphic. This symmetry type consists of a single isomorphy class for $m=3.$ For $m=2$, we would add intervals. We work in ${\mathbb{Q}}^{2},$ and no extension of the rational numbers is required. When we go to $m=4,$ we would obtain the parallelogram, the triangle, and a lot of more fragmented tiles [3,19].

## 9. Lattices from Polynomials

**Parameter-free description of mappings.**In this paper, a lattice was defined by a base $B=\{{b}_{1},{b}_{2}\}$ for which matrices $M,{s}_{k}$ have rational entries, and vectors ${v}_{k}$ have integer coordinates. How do we find B? Let us first reformulate condition (6) for a rotational symmetry and expansion.

**Proposition**

**8.**

- (i)
- The characteristic polynomial of s is ${p}_{s}\left(z\right)={z}^{2}+az+1,$ where a is rational and $\left|a\right|\le 2.$
- (ii)
- There are rational numbers $b,c$ such that $g\left(x\right)=bs\left(x\right)+cx$ for all $x.$

**Proof.**

**Companion matrix and exchange matrix.**The definition of a fractal family by one characteristic polynomial ${p}_{s}\left(z\right)$ and a polynomial equation $g={\sum}_{k=0}^{n}{b}_{k}{s}^{k}$ was implemented in IFStile for a higher-dimensional setting. The user has only to think about the master equations, while all matrix calculations are performed by the computer. In the case when the polynomial ${p}_{s}\left(z\right)$ is irreducible, the canonical base B is given by the successive images ${b}_{k}={s}^{k-1}\left({b}_{1}\right),\phantom{\rule{4pt}{0ex}}k=2,\dots ,n$ of the first base vector ${b}_{1}.$ In other words, we work with the companion matrix of $s.$ An advantage is that the transform to standard coordinates can be done by fast numerical procedures.

## 10. Constructions in Quadratic Number Fields

**Quadratic number fields.**Every irreducible polynomial p with rational coefficients gives rise to a field extension of $\mathbb{Q}.$ In particular, $p\left(z\right)={z}^{2}+d$ with a positive square-free integer d generates the field

**Proposition**

**9.**

- (i)
- ${p}_{s}$ generates the quadratic number field $\mathbb{Q}\left(\sqrt{-d}\right)$ where d is the square-free part of ${w}^{2}-{u}^{2}.$
- (ii)
- If a is different from 0 and 1, the rotation angle α is irrational and fulfils $cos\alpha =-a/2.$
- (iii)
- The parameter a corresponds to the integer triple $(u,v,w)$ which solves the equation ${u}^{2}+d{v}^{2}={w}^{2}.$ For every $d,$ these generalized Pythagorean triples are generated by Euclid’s formula$$u={n}^{2}-d{m}^{2}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}v=2mn\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}w={n}^{2}+d{m}^{2}\phantom{\rule{1.em}{0ex}}\mathit{with}\mathit{integers}m,n\mathit{such}\mathit{that}n>m\sqrt{d}.$$

**Proof.**

**Results of the computer experiments.**We studied rational numbers a with small denominator and $\left|a\right|<2,$ and the rotation with polynomial ${p}_{s}\left(z\right)={z}^{2}+az+1.$ The expansion was defined as $g=bs+c$ with rational numbers. The best results were obtained when g represents an algebraic integer in $\mathbb{Q}\left(\sqrt{-d}\right).$ That is, the determinant and trace of M in (10) must be integers. Recall that an element of the field (11) is an algebraic integer if either ${c}_{1},{c}_{2}$ are integers, or $d=3\mathrm{mod}4$ and ${c}_{1}={d}_{1}/2,\phantom{\rule{4pt}{0ex}}{c}_{2}={d}_{2}/2$ with odd integers ${d}_{1},{d}_{2}.$

**Summary.**Including irrational rotations in IFS is not easy. No tiles could be constructed this way. Nevertheless, we found connected self-similar sets with pieces intersecting in uncountable sets in all quadratic fields which we studied. Various examples have the topology of the Sierpiński carpet, almost the same Hausdorff dimension, and a much more interesting isotropic geometry. For the Sierpiński triangle, with finite intersections of pieces and dimension between 1.5 and 1.6, there are lots of variations in all quadratic fields considered. In general, the complexity of irrational examples is smaller than that of IFS with crystallographic data. Some of the examples with small numbers of neighbors seem to be unique and worth further study.

**Outlook.**This is a beginning. We discussed self-similar sets in the plane with equal contraction factors, between 4 and 14 pieces, and data from quadratic number fields. Graph-directed constructions and projection schemes for arbitrary number fields are more exciting. For three-dimensional self-similar sets, a general framework does not yet exist, despite much work, including [3,26,29,38,40,42,47].

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**Top**): Sierpiński triangle, Sierpiński carpet and a variation of Gosper’s snowflake tile are classical shapes with a simple open set. (

**Middle**): an example with crystallographic data and high complexity. The six pieces have considerable overlap and OSC is still true. Several magnifications are necessary to reveal the local structure. (

**Bottom**): a carpet with low complexity generated from an irrational rotation. There are no characteristic directions. See Section 6.

**Figure 2.**A carpet generated from an irrational rotation and a reflection. It has 13 neighbor types and no characteristic directions. For details, see Section 6.

**Figure 4.**The neighbor graph of Figure 3 has a very simple structure.

**Figure 5.**(

**Left**): second level pieces Figure 3, colored with respect to the second index. (

**Right**): the five possible neighbors of A determine the boundary sets.

**Figure 6.**A pattern of dimension 1.69 generated from $(3,4,5)$ together with ${90}^{\circ}$ rotations. Such patterns are fairly easy to generate. This one has only 9 neighbor types.

**Figure 7.**A carpet generated from $(3,4,5)$ together with ${90}^{\circ}$ rotations. The expansion has determinant 10, and there are $m=8$ pieces. The Hausdorff dimension is 1.81, the intersections of pieces have dimension 1.25, and there are 13 neighbor types.

**Figure 8.**A carpet generated from irrational rotation by $\mathrm{arctan}\phantom{\rule{0.166667em}{0ex}}\frac{3}{13}$ together with ${60}^{\circ}$ rotations. The expansion has determinant 7, so for $m=6$ pieces we have dimension 1.84. There are 18 neighbor types.

**Figure 9.**A carpet generated from three irrational rotations with $a=\frac{3}{2},\frac{1}{4},\frac{9}{8}$ and an expansion g of determinant of 8. It has dimension 1.87 and 19 neighbor types. The number field is $\mathbb{Q}\left(\sqrt{-7}\right),$ as for the tame twindragon.

**Figure 10.**One of the simplest carpets for $a=1/2,$ which corresponds to $\mathbb{Q}\left(\sqrt{-15}\right)$.

**Table 1.**The first positive square-free integers $d,$ and corresponding fractions a with denominators up to 9 which generate the field $\mathbb{Q}\left(\sqrt{-d}\right)$.

d | a | d | a | d | a |
---|---|---|---|---|---|

1 | $0,\phantom{\rule{4pt}{0ex}}\frac{6}{5},\phantom{\rule{4pt}{0ex}}\frac{8}{5}$ | 10 | $\frac{6}{7}$ | 19 | $\frac{9}{5}$ |

2 | $\frac{2}{3},\phantom{\rule{4pt}{0ex}}\frac{14}{9}$ | 11 | $\frac{5}{3},\phantom{\rule{4pt}{0ex}}\frac{1}{5},\phantom{\rule{4pt}{0ex}}\frac{7}{9}$ | 21 | $\frac{4}{5}$ |

3 | $1,\frac{2}{7},\frac{11}{7},\frac{13}{7}$ | 13 | $\frac{12}{7}$ | 23 | $\frac{11}{6},\phantom{\rule{4pt}{0ex}}\frac{7}{8}$ |

5 | $\frac{4}{3},\phantom{\rule{4pt}{0ex}}\frac{4}{7},\phantom{\rule{4pt}{0ex}}\frac{2}{9}$ | 14 | $\frac{10}{9}$ | 31 | $\frac{15}{8}$ |

6 | $\phantom{\rule{4pt}{0ex}}\frac{2}{5},\phantom{\rule{4pt}{0ex}}\frac{10}{7}$ | 15 | $\frac{1}{2},\phantom{\rule{4pt}{0ex}}\frac{7}{4},\phantom{\rule{4pt}{0ex}}\frac{11}{8}$ | 33 | $\frac{8}{7}$ |

7 | $\frac{3}{2},\phantom{\rule{4pt}{0ex}}\frac{7}{4},\phantom{\rule{4pt}{0ex}}\frac{9}{8}$ | 17 | $\frac{16}{9}$ | 35 | $\frac{1}{3},\phantom{\rule{4pt}{0ex}}\frac{17}{9}$ |

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**MDPI and ACS Style**

Bandt, C.; Mekhontsev, D.
Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations. *Fractal Fract.* **2022**, *6*, 39.
https://doi.org/10.3390/fractalfract6010039

**AMA Style**

Bandt C, Mekhontsev D.
Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations. *Fractal and Fractional*. 2022; 6(1):39.
https://doi.org/10.3390/fractalfract6010039

**Chicago/Turabian Style**

Bandt, Christoph, and Dmitry Mekhontsev.
2022. "Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations" *Fractal and Fractional* 6, no. 1: 39.
https://doi.org/10.3390/fractalfract6010039