A Brief Survey of Paradigmatic Fractals from a Topological Perspective

Abstract

The key issues in fractal geometry concern scale invariance (self-similarity or self-affinity) and the notion of a fractal dimension $D$ which exceeds the topological dimension $d$. In this regard, we point out that the constitutive inequality $D>d$ can have either a geometric or topological origin, or both. The main topological features of fractals are their connectedness, connectivity, ramification, and loopiness. We argue that these features can be specified by six basic dimension numbers which are generally independent from each other. However, for many kinds of fractals, the number of independent dimensions may be reduced due to the peculiarities of specific kinds of fractals. Accordingly, we survey the paradigmatic fractals from a topological perspective. Some challenging points are outlined.

1. Introduction

Presently, the notion of fractals is widely used in sciences and arts [1,2,3]. Although the term fractal was introduced by Benoit Mandelbrot in 1975 [4], most classical fractals were already created a long time ago. Namely, in 1872, Karl Weierstrass [5] presented an everywhere continuous but nowhere differentiable function, now known as the Weierstrass function. The graphs of this function are often used to illustrate the concept of self-affine fractals [6]. In 1983, Georg Cantor [7] studied a set of points lying on a single line segment that has many unintuitive properties. Although this set was earlier explored by Henry John Stephen Smith [8], it was named as the Cantor set since through the consideration of this set Cantor helped to lay the foundations of modern point-set topology [9]. In 1890 Giuseppe Peano [10] discovered the first example of space-filling curves, now called the Peano curve [11]. Immediate1y after that, David Hilbert [12] replaced the purely arithmetic definition proposed by Peano with a more descriptive geometric construction of graph and built up a variant of the Peano curve now referred to as the Hilbert curve [13]. The most famous space-filling curve called the Sierpiński square snowflake [14] was invented by Wacław Sierpiński [15]. Sometime before, Niels Fabian Helge von Koch [16] devised the simplest variant of a continuous curve without tangents, which is now called the Koch curve [17]. In 1915, Sierpiński [18] created a Koch-like curve, forming a self-similar shape later identified as the Sierpiński gasket [19]. One year later, Sierpiński [20] constructed a curve forming an infinitely ramified network, which is now known as the Sierpiński carpet [21]. An analogous curve in three dimensions was presented by Karl Menger in 1926 [22]. The Sierpiński carpet and the Menger sponge can be viewed as analogs of the Cantor set on the plane and in three-dimensional space, respectively [23]. In 1957, Simon Broadbent and John Hammersley [24] developed the first mathematical model of percolation. It is pertinent to note that the scaling properties of percolation were established before the fractal nature of critical percolation clusters was recognized [25].

In this regard, Felix Hausdorff noted that the standard topological definition was not suitable for some sets. Consequently, he introduced a new definition of dimension based on the set size variations with the scale of measurements [26]. Formally, the Hausdorff dimension is defined as

$$D_H=\inf \left\{s\ge 0:{\mathcal{H}}_H^s=0\right\}=\sup \left\{s\ge 0:{\mathcal{H}}_H^s=\infty \right\},$$

(1)

where

$${\mathcal{H}}_H^s=\underset{\epsilon \to 0}{\lim }\inf \left\{\sum _{i\in I}\mathrm{diam}{\left(U_i\right)}^s:{\left\{U_i\right\}}_{i\in I}\in {\mathcal{C}}_{\epsilon }\right\}$$

(2)

is the Hausdorff measure, while ${\mathcal{C}}_{\epsilon }$ is the collection of all $\epsilon$-covers of the studied object. Although the definitions (1) and (2) is based on the earlier ideas of Carathéodory [27], Hausdorff first realized that this construction allows the definition of fractional dimension numbers. In particular, he proved that the middle-third Cantor set is characterized by the fractional dimension $D_H=\ln 2/\ln 3$. Inherent features and technical aspects regarding the Hausdorff measure and dimension were described by Besicovitch [28,29].

Difficulties with implementing the Hausdorff dimension for numerical applications motivate the development of empirical estimates. Nowadays, the most used dimension number is the box-counting dimension based on the concept of “measurement at scale $\epsilon$” [30]. Accordingly, the box-counting dimension can be easily computed as

$$D_B=-\underset{\epsilon \to 0}{\lim }\left[\ln N\left(\mathsf{\epsilon }\right)/\ln \epsilon \right]$$

(3)

if this limit exists, while $N\left(\epsilon \right)$ is the smallest number of boxes of size $\epsilon$ (or at most $\epsilon$) needed to cover the studied pattern. If the limit does not exist, one can employ $\underset{\epsilon \to 0}{\lim }\inf$ or $\underset{\epsilon \to 0}{\lim }\sup$. Moreover, there are many variants of definition for the box-counting dimension (e.g., capacity dimension, Minkowski dimension, etc.) [31]. The differences between them mainly concern the shape of boxes used for pattern covering. Nonetheless, it has been proved that, generally, $D_B\ge D_H$. The popularity of the box-counting dimension is mainly due to its robustness in the Euclidean context [32].

In this background, Benoit Mandelbrot published his celebrated paper “How long is the coast of Britain? Statistical self-similarity and fractional dimension”, in which he resolved the Steinhaus Paradox that the measured length of geographic features (such as coasts and rivers) increases with increasing map scales [33]. Sometime afterward, Mandelbrot [1] created the word fractal as a reference to patterns that are fragmented, not smooth and irregular. In 1977, Mandelbrot [34] coined the notion of fractals to define a large class of mathematical and natural objects that share the property of self-similarity or self-affinity. In a unified way, the self-similar and self-affine sets were presented by John Hutchinson through the theory of the Iterated Function System (IFS) [35]. The IFS is defined as a finite set of contraction mappings. When the mappings are affine transformations, the IFS attractor tends to be self-affine or self-similar. Accordingly, the scale invariance is characterized by the similarity dimension $D_S$ defined by the following relation:

$${\sum }_{i=1}^m{\lambda }_i^{D_S}=1,$$

(4)

where ${\lambda }_i$ are the ratios of contraction (${0<\lambda }_i<1$), and $m$ is the number of contractions in the iterated function system ${\left\{S_i\right\}}_{i=1}^m$ [36]. It has been recognized that the similarity and Hausdorff dimensions are equivalent when the IFS satisfies the open set condition [31].

Although there is no canonical definition of fractals, the term fractal is commonly used to refer to a self-similar shape whose fractal dimension $D$ is larger than its topological dimension [32]. The role of fractal dimension $D$ can play any suitable defined dimension number associated with the fractal’s ability to fill the embedding Euclidean space [37,38,39,40]. Specifically, the Hausdorff (1), (2), box-counting (3), similarity (4), packing [31], and Assouad [41] dimensions are frequently used. However, fractals having the same values of $D$ may have very different topological, metrological, morphological, and topographical properties [42,43,44,45]. Therefore, in order to unambiguously characterize the inherent features of fractals, one needs to employ more dimension numbers, though yet there is no consensus about the number and identities of basic dimension numbers (see, for review, Refs. [46,47,48,49,50] and references therein).

In this work, we argue that the main topological features of fractals can be specified by six generally independent dimension numbers: the topological dimension $d$, the fractal dimension $D$, the topological fractal dimension $D_{tF}$, the connectivity (chemical, spreading) dimension $d_{\mathcal{l}}$, the spectral dimension $d_s$, and the fractal dimension of the random walk $D_W$. Other dimension numbers used to characterize the topological and metrological properties of fractals can be expressed in terms of these six dimensions. Furthermore, for many types of fractals, there are specific relations between the basic dimension numbers, which are different for distinct types of fractals. Accordingly, in this work, we survey paradigmatic fractals from a topological perspective. The rest of the paper is organized as follows. Section 2 is devoted to the classifications of fractals and the definitions of dimension numbers. The topological features of fractals are scrutinized. A set of generally independent dimension numbers is suggested. Section 3 is devoted to a survey of paradigmatic fractals from the topological perspective. In this way, some new relations are established. Several challenging points are outlined. The main conclusions are summarized in Section 4.

2. Key Dimension Numbers and the Classification of Fractals

Two main issues in fractal geometry are the scale-invariance (self-similarity or self-affinity) and the notion of fractal dimension. The scale-invariance implies that the measure $M(L/\epsilon )$ associated with the fractal dimension $D$ scales as

$$M\left[\lambda \left(L/\epsilon \right)\right]\propto {\lambda }^{-D}M\left(L/\epsilon \right),$$

(5)

where $L$ is the pattern size, $\epsilon$ is the resolution length (e.g., box size), and $\lambda >0$, while the inequality

$$D>d$$

(6)

holds per the definition of fractals [32]. For self-similar fractals, $\lambda$ is the same for different directions, whereas the self-affine fractals exhibit different rates of contraction ${\lambda }_i$ in different directions $i=\mathrm{1,2},\dots n$. The scaling anisotropy is commonly characterized by the so-called Hurst exponent, defined as

$$H=\ln {\lambda }_i/\ln {\lambda }_j,$$

(7)

where $0<{\lambda }_j\le {\lambda }_i\le 1$, $j\ne i$ [51]. It was also demonstrated that the local box-counting dimension of the self-affine fractal is equal to

$$D_B=d+1-H,$$

(8)

whenever $0<H<1$, whereas for self-similar fractals, ${\lambda }_j={\lambda }_i$, and so $H=1$. However, the length of the self-affine curve increases with a decrease in the resolution length $\epsilon \to 0$ as $\mathcal{L}\propto {\epsilon }^{1-D_D}$, where $D_D$ is the local divider (compass) dimension [51] equal to

$$D_D=1/H, \mathrm{if} 0<H<0.5, \mathrm{or} D_D=2, \mathrm{if} 1>H\ge 0.5,$$

(9)

and so, for self-affine patterns, $D_D>D_B$.

From a topological perspective, the fractal can be totally disconnected (e.g., Cantor set and dust shown in Figure 1a,b, respectively), disconnected (e.g., Cantor circles, in Figure 1c, or connected and path-connected (e.g., Cantor tartan in Figure 1d). We recall that a fractal is connected if it cannot be decomposed into the union of two nonempty, disjoint, closed sets. If such decomposition exists, then the fractal is said to be disconnected. If the connected component of every point of the fractal is only the point itself, then the fractal is totally disconnected. The connected fractal is called path-connected if along with any pair of points, it also contains a path connecting these points [52]. Disconnected and connected fractals can be finitely or infinitely ramified and can be loopless or contain loops at all scales. Accordingly, a fractal topology must be characterized by the connectedness, the connectivity, the ramification, and the loopiness.

In contrast to the fractal dimension associated with a suitable defined measure, the topological dimension is associated with how a pattern can be separated into parts. It is defined recursively as follows. A pattern $F$ has the topological dimension $d$ if it can be divided by an object having the topological dimension $d-1$, while a point has the topological $d=0$ per definition [40]. Formally, this can be expressed as

$$d\left(F\right)=\min \left\{s:\exists X\subseteq F \mathrm{such} \mathrm{that} d\left(F\right)\le s-1 \mathrm{and} d(F\backslash X)\le 0\right\},$$

(10)

where the topological dimension of empty space is assumed to be $d=-1$. Accordingly, the topological dimension takes only integer values [53].

Fractals having the topological dimension $d\ge 1$ can be finitely or infinitely ramified. The ramification of the fractal pattern can be quantified by the topological Hausdorff dimension [54] which was defined in Ref. [53] via the combination of Equations (1), (2), and (10) as follows:

$$D_{tH}\left(F\right)=\min \left\{s:\exists X\subseteq F \mathrm{such} \mathrm{that} D_H\left(F\right)\le s-1 \mathrm{and} d(F\backslash X)\le 0\right\},$$

(11)

where $s$ is a real number larger than or equal to zero. Later, the definition (7) was generalized to define the topological fractal dimension as

$$D_{tF}\left(F\right)=\min \left\{s:\exists X\subseteq F \mathrm{such} \mathrm{that} D\left(F\right)\le s-1 \mathrm{and} d(F\backslash X)\le 0\right\},$$

(12)

where $D\left(F\right)$ can be the box-counting dimension (3) or the similarity dimension (4) [55,56,57]. The totally disconnected fractals are characterized by $D>D_{tF}=d=0$, while for the path-connected fractals, $D\ge D_{tF}\ge d\ge 1$, whereas the disconnected fractals have $D\ge D_{tF}>d\ge 1$. It was also recognized that the topological fractal dimension is equal to

$$D_{tF}=Q+1,$$

(13)

where $Q$ is the ramification exponent [45]. Accordingly, the finitely ramified patterns (see, for example, Figure 2 and Figure 3a–c) have $D_{tF}=d=1$, whereas the infinitely ramified fractals (see Figure 1c,d and Figure 3d) are characterized by $D_{tF}>1$.

The connectivity of the path-connected fractal is characterized by the connectivity dimension, defined as

$$d_{\mathcal{l}}=-\underset{\mathcal{l}\to 0}{\lim }\left[\ln \mathcal{N}\left(\mathcal{l}\right)/\ln \mathcal{l}\right],$$

(14)

where $\mathcal{N}\left(\mathcal{l}\right)$ is the number of points connected with an arbitrary point inside of a $d_{\mathcal{l}}$-dimensional ball of radius $\mathcal{l}$ around this point [45]. In the literature, the connectivity dimension is sometimes called the chemical dimension, spreading dimension, and growth exponent [58,59,60]. In contrast to the box-counting dimension (3), which describes how the fractal measure scales with its Euclidean size, the connectivity dimension (14) characterizes the internal connectivity between the elements of the fractal structure. Accordingly, the ratio of the fractal dimension to the connectivity dimension is equal to the fractal dimension of the minimum path:

$$d_{\min }=D/d_{\mathcal{l}},$$

(15)

which characterizes the scaling behavior of the length of the shortest path between two arbitrary chosen points on the fractal $\mathcal{l}\propto r^{d_{\min }}$, where $r$ is the Euclidean distance between these points.

In this regard, we noted that the constitutive inequality (6) can have either a geometric or topological origin, or both. A pure geometrical origin of the inequality (6) implies that

$${D>d}_{\mathcal{l}}=D_{tF}=d\ge 1,$$

(16)

as this is for self-avoiding self-similar (self-affine) curves (see Figure 2a–c) and surfaces (e.g., Cartesian products of self-avoiding Koch curves with a straight line), whereas a pure topological origin implies that

$$d_{\mathcal{l}}=D\ge D_{tF}\ge d\ge 1,$$

(17)

as this is for the Vicsek fractal shown in Figure 3a, the Sierpiński gasket and pyramid shown in Figure 3b,c, and the Sierpiński carpet shown in Figure 3d. Notice that the fractals obeying the relations (17) can be finitely or infinitely ramified. The finitely ramified fractals with a pure topological origin of the inequality (6) form the class of post-critically finite (PCF) fractals defined by Kigami as the finite ramified self-similar sets which allow a pure topological description [61]. It is a straightforward matter to understand that the PCF fractals (e.g., the branched Koch curve shown in Figure 2d, Vicsek fractals, and Sierpiński) obey more restricted relations,

$$d_{\mathcal{l}}=D>D_{tF}=d=1,$$

(18)

whereas the infinitely ramified fractals are characterized by $D_{tF}>1$. Notice that the class of PCF fractals includes the subclass of so-called nested fractals (e.g., Vicsek fractals and Sierpiński gaskets) introduced axiomatically by Lindstrom [62].

In the case of a mixed origin of the constitutive inequality (6), the dimension numbers obey the following relations:

$$D>D_{tF}\ge d\ge 1 \mathrm{and} {D>d}_{\mathcal{l}},$$

(19)

as this is for the diamond fractal shown in Figure 4 and for critical percolation clusters (see Ref. [55]).

The connectivity dimension also controls a relation between the number of effective spatial and dynamical degrees of freedom of a random walker on the fractal. Specifically, in Ref. [63], it has been established that

$$n_{\gamma }+d_s=2d_{\mathcal{l}},$$

(20)

where the number of effective spatial degrees of freedom $n_{\gamma }$ can be viewed as the number of directions determined by the topological constraint, while the number of effective dynamical degrees of freedom is equal to the spectral dimension $d_s$. The spectral dimension governs the density of vibrational states [64,65]. Consequently, it is equal to the Lagrangian dimension [66], and so $d_s$ determines the scaling properties of the eigenvalues of the Laplacian defined on the fractal [67]. Accordingly, the spectral dimension can be calculated using the following scaling relation:

$$d_s=-2\partial \ln 〈P\left(\tau \right)〉/\partial \ln \tau ,$$

(21)

where $P\left(\tau \right)$ is the probability that a random walker on the fractal returns to its origin after $\tau$ steps, and $〈\dots 〉$ denotes the ensemble average [45].

The generalized Einstein law connecting the electrical conductivity to the extensive diffusion coefficient implies that the electrical resistance exponent $\zeta$ is related to the fractal dimension of random walk $D_W$ as

$$\zeta =D_W\left(1-d_s/2\right),$$

(22)

while $D_W$ is defined via the scaling behavior of the mean squared displacement of the random walker $〈r^2〉\propto {\tau }^{2/D_W}$ [68]. For the loopless fractals, $\zeta =d_{\min }$, whereas for the fractals with loops at all scales, $\zeta <d_{\min }$, and $\zeta$ is negative if $d_s>2$. Quantitatively, the fractal loopiness can be characterized by the loopiness index:

$$\Lambda =d_s/n_{\gamma }-1/d_{\mathcal{l}},$$

(23)

which can vary in the range of $0\le \Lambda \le 1$, while for the loopless fractals, $\Lambda =0$ [45].

Thus, the main topological properties of path-connected fractals can be characterized by six dimension numbers which are generally independent. Specifically, we suggest the following set of basic dimension numbers: the topological dimension $d$, the fractal dimension $D$, the topological fractal dimension $D_{tF}$, the connectivity (chemical, spreading) dimension $d_{\mathcal{l}}$, the spectral dimension $d_s$, and the fractal dimension of the random walk $D_W$. Other dimension numbers used to characterize the topological and metrological properties of fractals can be expressed in terms of these six dimensions. Furthermore, many kinds of fractals obey the Alexander–Orbach relation:

$$d_s=2D/D_W=2d_{\mathcal{l}}/d_W,$$

(24)

where $d_W=D_W/d_{\min }$ [46]. For the path-connected loopless fractals,

$$D_W=D+d_{\min },$$

(25)

whereas for fractals having loops at all scales, the fractal dimension of the random walk is in the range of

$${2\le D}_W=D+\zeta <D+d_{\min }$$

(26)

Accordingly, the random walks on path-connected fractals serve as a paradigm of anomalous subdiffusion with $D_W>2$ [69], whereas the anomalous superdiffusion can be modeled by the random walks on the totally disconnected fractals [70].

3. Survey of Paradigmatic Fractals from Topological Perspectives

3.1. Self-Avoiding Koch Curves

The first example of a self-similar, self-avoiding curve on a plane is the triadic Koch curve, the iterative construction of which is shown in Figure 2a. This curve is commonly used to illustrate a notion of a nonrectifiable curve, as well as an example of a continuous but nowhere differentiable function [71]. The triadic Koch curve may be also obtained as the invariant attractor set of the IFS having four similarity transformations with the reduction ratio 1/3 [72]. On the other hand, the Koch curve can be viewed as an analog of the fractal interpolation theorem of Barnsley [73]. It was also shown that there exists a homeomorphism between the closed interval $\left[\mathrm{0,1}\right]\subset \mathbb{R}$ and the Koch curve endowed with the subset topology of ${\mathbb{R}}^2$ [74]. The fractal and nonfractal properties of the triadic Koch curve were discussed in Refs. [75,76,77,78,79,80]. The generalizations of the Koch curve were discussed in Refs. [81,82,83]. A most famous variant of the Koch curve, known as the Minkowski sausage, is shown in Figure 2b. It has the fractal dimension $D=ln8/ln4=1.5$. Figure 2c shows the self-affine Koch curve ($H=1/2$) with the same box-counting dimension as the Minkowski sausage, whereas its divider (compass) dimension is equal to $D_D=2$. Generally, the curve constructed by the Koch algorithm can intersect itself and so form a network with loops [84]. The conditions of self-avoidance within the Koch algorithm were discussed in [85].

From the topological viewpoint, the self-avoiding Koch curves are path-connected, finitely ramified, loopless fractals. For such fractals, the inequality (6) has a purely geometric origin. Consequently, the dimension numbers of the self-avoiding fractal curve obey relations (16) with $d=1$, such that $D=d_{\min }$, $D_W=2d_{\min }>2$, and $d_s=1$.

The differential equation describing the subdiffusion of the self-avoiding Koch curve was derived in [86]. On the other hand, it was recognized that long-range interactions between the points on the Koch curve can change its spectral properties [87]. In particular, it was found that if the interactions decay with the distance $r$ as a power law $p\propto r^{-\alpha }$, then the spectral dimension of the triadic Koch curve depends on the interaction exponent $\alpha$ as

$$d_s=\left\{\begin{array}{cc}1,& \mathrm{if}\alpha \ge \ln 4/\ln 3,\\ 2\ln 4/\left(\ln 4+\alpha \ln 3\right),& \mathrm{if} \ln 2/\ln 3<\alpha <\ln 4/\ln 3,\\ 4/3,& \mathrm{if}\alpha \le \ln 2/\ln 3,\end{array}\right.$$

(27)

while the fractal (similarity, box-counting) dimension $D=\ln 4/\ln 3$ is obviously independent of interactions. This finding can be interpreted as an increase in fractal connectivity as the interaction exponent decreases. Accordingly, from Equation (27), it follows that the connectivity dimension increases with the decrease $\alpha$ as

$$d_{\mathcal{l}}=\left\{\begin{array}{cc}1,& \mathrm{if}\alpha \ge \ln 4/\ln 3,\\ \ln 4/\left(\alpha \ln 3\right),& \mathrm{if} \ln 2/\ln 3<\alpha <\ln 4/\ln 3,\\ 2,& \mathrm{if}\alpha \le \ln 2/\ln 3,\end{array}\right.$$

(28)

such that the electrical resistance exponent is $\zeta =d_{\min }=D$ $\mathrm{if}\alpha \ge \ln 4/\ln 3$, $\zeta =\alpha$ $\mathrm{if} \ln 2/\ln 3<\alpha <\ln 4/\ln 3$, and $\zeta =D/2$ $\mathrm{if}\alpha \le \ln 2/\ln 3$.

Equations (27) and (28) can be easily generalized for self-avoiding Koch curves with any fractal dimension in the range of $1<D<2$. In this way, we find that

$$d_{\mathcal{l}}=\left\{\begin{array}{cc}1,& \mathrm{if}\alpha \ge D,\\ D/\alpha ,& \mathrm{if} D/2<\alpha <D,\\ 2,& \mathrm{if}\alpha \le D/2,\end{array}\right.$$

(29)

such that the spectral dimension is equal to

$$d_s=\left\{\begin{array}{cc}1,& \mathrm{if}\alpha \ge D,\\ 2D/\left(D+\alpha \right),& \mathrm{if} D/2<\alpha <D,\\ 4/3,& \mathrm{if}\alpha \le D/2,\end{array}\right.$$

(30)

while the electrical resistance exponent is equal to $\zeta =D/d_{\mathcal{l}}$.

From Equations (29) and (30) together with Equation (20), it follows that the loopiness index (23) is equal to $\Lambda =0$ independent of the long-range interactions, that is, the curve with the long-range interactions remains loopless. Nonetheless, the long-range interactions can lead to superdiffusion with the random walk dimension:

$$D_W=D+\zeta =D+D/d_{\mathcal{l}}<2,$$

(31)

if the fractal dimension of the Koch curve is in the range of $1<D<4/3$, such that $D/2<2-D$.

3.2. Sierpiński Gasket Shape (Network versus Self-Avoiding Curve) and PCF Fractals

The Sierpiński gasket is probably the most studied deterministic fractal. In fact, the analysis on fractals was originated by the construction of the Brownian motion on the Sierpiński gasket as a scaling limit of random walks on prefractal Sierpiński networks [88,89,90]. Since then, the Sierpiński gasket has given rise to the concept of the resistance metric [91] and to the notions of PCF fractals [61], nested fractals [62], the fundamental group of the Sierpiński gasket [92], and topological fractals [93].

In this regard, it is pertinent to note that the aim of the original Sierpiński paper [18] was to construct an example of a Cantorian curve. Accordingly, the Sierpiński gasket was created by an iterative process analogous to the construction of self-avoiding Koch curves, as shown in Figure 5a. Although at every iteration this construction remains a self-avoiding curve, in the limit of infinite iterations, the arrowhead curve traces out the Sierpinski triangle by a single continuous directed path. Alternatively, the Sierpiński gasket can be constructed by the iterative procedure shown in Figure 5b. Furthermore, there are many other IFS systems which have the Sierpiński gasket shape as the attractor [94,95,96,97,98]. Though, despite the same geometric shape being characterized by the fractal dimension $D=\ln 3/\ln 2$, the topological properties of the fractal obtained as the limit arrowhead curve differ from those for the gasket obtained in the limit of the Sierpiński network. This was explicitly recognized from the comparison of the Laplacians constructed as the scaling limit of Laplacians on prefractal arrowhead curves and on the prefractal Sierpiński network [99]. Namely, in this way, it was found that the arrowhead curve has the spectral dimension $d_s=1$, whereas the spectral dimension of the Sierpiński network is $d_s=2\ln 3/\ln 5$ [100]. Furthermore, it is straightforward to verify that the arrowhead curve obeys the relations (16) for self-avoiding, self-similar curves, whereas the Sierpiński network (gasket) obeys the relations (18) for nested PCF fractals.

The Sierpiński gasket on a plane ($n=2$) can be easily generalized to the Sierpiński pyramids in n-dimensions ($n\ge 2$). In this case, the number of contractions is equal to $n+1$, while the contraction ratio is $\lambda =l_{i+1}/l_i=1/2$, as for example in Figure 3b,c. Accordingly, the fractal and spectral dimensions are equal to

$$D=\ln \left(n+1\right)/\mathrm{l}\mathrm{n}\left(2\right) \mathrm{and} d_s=2\ln \left(n+1\right)/\mathrm{l}\mathrm{n}\left(\mathrm{n}+3\right),$$

(32)

respectively [101], while other dimension numbers obey the standard relations for the PCF fractals (18) with $d=1$. Notice that as the embedding dimension $n\to \infty$, the spectral dimension $d_s\to 2$, whereas the fractal dimension $D\to \infty$, and the loopines index $\Lambda \to 0$. Another generalization of the Sierpiński gasket on a plane is associated with changes in the contraction ratio (see Figure 6). In this case, the similarity dimension is equal to

$$D=\mathrm{l}\mathrm{n}\left[{\lambda }^{-1}\left({\lambda }^{-1}+1\right)/2\right]/\mathrm{l}\mathrm{n}\left({\mathsf{\lambda }}^{-1}\right),$$

(33)

such that $D\to 2$ as $\lambda \to 0$. Although an analytic expression for the spectral dimension as a function of $\lambda$ is unknown, it was established that $d_s\to 2$ as $\lambda \to 0$, and so $\Lambda \to 0.5$. In the particular case of $\lambda =3$, the spectral dimension is found to be $d_s=2\ln \left(6\right)/\mathrm{l}\mathrm{n}\left(90/7\right)$ [101]. The peculiar properties of Sierpiński gaskets remain an active topic of research up to now (see, for instance, Refs. [102,103,104,105,106,107,108,109,110] and references therein).

Another family of nested PCF fractals is represented by the loopless Vicsek fractals. Originally, these fractals were introduced by Vicsek in 1989 [111]. The iterative construction of a Vicsek fractal with the coordination number $f=4$ is shown in Figure 3a. Generally, the fractal dimension of the Vicsek fractal depends on the coordination number $f$ and the contraction ratio $\lambda$ as

$$D=-\ln \left(f+1\right)/\ln (\lambda ),$$

(34)

while other dimension numbers obey the relations (18) for loopless PCF fractals. Accordingly, the spectral dimension of a Vicsek fractal is equal to

$$d_s=2D/\left(D+1\right)=2\ln \left(f+1\right)/\ln \left({\lambda }^{-1}f+{\lambda }^{-1}\right),$$

(35)

while the loopiness index is $\Lambda =0$. The properties and applications of Vicsek fractals were discussed in Refs. [112,113,114,115].

3.3. The Standard Sierpiński Carpets

Sierpiński [20] proposed to construct self-similar patterns recursively by removing the interior of a middle square in each step, ad infinitum. The first three steps of the construction of the classic Sierpiński carpet $S_3^1$ are shown in Figure 3d. A generalization of the Sierpiński algorithm has given rise to the family of fractal squares which includes connected, disconnected, and totally disconnected fractals (see, for instance, Refs. [116,117,118,119,120]). The fractal square is obtained by subdividing a square into $N$ subsquares of equal size and removing $M\ge 1$ of them at each iteration step. Accordingly, the fractal dimension of the fractal square is equal to

$$D=2\ln \left(N-M\right)/\mathrm{l}\mathrm{n}\left(N\right),$$

(36)

independent of the positions of the removed subsquares. In contrast to this, the connectedness, connectivity, loopiness, and ramification are strongly dependent on the positions of the removed subsquares (see, for details, Refs. [121,122,123,124,125,126,127,128]).

A standard Sierpiński carpet $S_{\sqrt{N}}^{\sqrt{M}}$ can be constructed in a similar way by dividing the unit square into $N=n\times n$ subsquares of equal edge size $1/\sqrt{N}$ and removing the interior of the central square element containing $M=m\times m$ subsquares, with $m=2i\le n-2$ ($i=\mathrm{1,2},3\dots$) if $n\ge 4$ is even, or $m=2i-1\le n-2$ if $n\ge 3$ is odd (see, for example, Figure 7). This procedure is recursively repeated with each of $\left(N-M\right)$ remaining subsquares, ad infinitum.

The standard Sierpiński carpets display many fascinating topological properties that appear in a vast variety of contexts [129,130,131,132,133,134,135,136,137,138,139,140,141]. It was found that the degree (averaged coordination number) of the prefractal carpets increases with the number of iterations $k$ as

$$Z_k=Z_{\infty }\left[1-N^{k\left(1-D\right)/2}\right],$$

(37)

while in the fractal limit of $k\to \infty$, the degree of the standard Sierpiński carpet is equal to

$$Z_{\infty }=4\left[\left(N-M-\sqrt{N}-\sqrt{M}\right)/\left(N-M-\sqrt{N}\right)\right],$$

(38)

and so $Z_{\infty }<4$ [139]. The carpet constrictivity can be quantified by the constrictivity factor $\delta$ defined as the averaged ratio between the measures (e.g., length or area) of consecutive different cross-sections [141]. It was found that the constrictivity factor of the standard Sierpiński carpet is equal to

$$\delta =1-\sqrt{M/N},$$

(39)

independent of the number of iterations. In Ref. [54], it was established that the topological fractal dimension of the standard Sierpiński carpet is equal to

$$D_{tF}=1+2\mathrm{l}\mathrm{n}\left(\sqrt{N}-\sqrt{M}\right)/\ln \left(N\right)=2\mathrm{l}\mathrm{n}\left(\delta N\right)/\ln \left(N\right).$$

(40)

Accordingly, the standard Sierpiński carpets obey the relations (17) with $d=1$, while $d_{\mathcal{l}}=D$.

It was demonstrated that the fractal dimension of the random walk can be estimated using the following relation:

$$D_W=2\ln \left(N+M\right)/\ln \left(N\right)=2+2\ln \left(1+M/N\right)/\ln \left(N\right),$$

(41)

which provides a good approximation to the data of numerical simulations on standard Sierpiński carpets [54]. It was established early that standard Sierpiński carpets obey the Alexander–Orbach relation (24) [142]. Accordingly, the spectral dimension of the standard Sierpiński carpet can be estimated with the following relationship:

$$d_s=2\ln \left(N-M\right)/\ln \left(N+M\right),$$

(42)

such that the standard Sierpiński carpets obey the following inequalities:

$${D_{tF}<d}_s<D,$$

(43)

whereas, generally, the spectral dimension can be larger than, equal to, or smaller than the topological fractal dimension [45].

Thus, taking into account the equality $d_{\mathcal{l}}=D$ and the Alexander–Orbach relation (24), the topological properties of the standard Sierpiński carpet can be characterized by four from six independent dimension numbers. However, in Ref. [141], it was argued that the scaling properties of connected fractal squares are characterized by two more dimension numbers—the mean fractal dimension of slices $〈D_i〉$ and the topological fractal dimension of the random walk $D_{tW}$. The former is defined as follows:

$$〈D_i〉=\underset{k\to \infty }{\lim }\left(N^{-k/2}{\sum }_{i=1}^{N^{k/2}}{D}_{i,k}\right),$$

(44)

where $D_{i,k}$ denotes the fractal dimension of slice $i$ in the prefractal carpet after $k$ iterations divided into $N^{k/2}$ slices [141]. The topological fractal dimension of the random walk is defined via the scaling relation

$${\varrho }_k=N^{k/2}{\left({\sum }_{i=1}^{N^{k/2}}{\varphi }_{i,k}^{-1}\right)}^{-1}\propto N^{-k\left(D_{tW}-D_{tH}\right)/2},$$

(45)

where ${\varphi }_{i,k}$ is the porosity of slice $i$ in the prefractal carpet after $k$ iterations. In Ref. [141], it was established that the topological fractal dimension of the standard Sierpiński carpet is equal to

$$D_{tW}=2\ln \left(\delta N+M\right)/\ln \left(N\right),$$

(46)

whereas values of $〈D_i〉$ for different fractals were obtained from numerical calculations. In the present work, we found that for standard Sierpiński carpets, the mean fractal dimension of slices can be expressed in terms of the topological fractal dimension (40) and the constrictivity factor (39) as follows:

$$D_{tW}=2\ln \left(\delta N+M\right)/\ln \left(N\right),$$

(47)

and so ${〈D_i〉>D}_{tF}$ because for standard Sierpiński carpets, $D_{tF}<D<2$. The dimension numbers characterizing the topological features of Sierpiński carpets shown in Figure 7 are summarized in

Table 1. Dimension numbers (basic: topological $d$, topological fractal $D_{tF}$, fractal $D$, connectivity $d_{\mathcal{l}}$, and spectral $d_s$ dimensions; additional: the mean fractal dimension of slices $〈D_i〉$ and the topological fractal dimension of the random walk $D_{tW}$) and dimensionless parameters (resistance exponent $\zeta$, loopiness index $\Lambda$, constrictivity factor $\delta$) characterizing the standard Sierpiński carpets $S_N^M$.

${\mathit{S}}_{\mathit{N}}^{\mathit{M}}$		Basic Dimension Numbers				Parameters			Dimension Numbers
$N$	$M$	$d$	$D_{tF}$	$D=d_{\mathcal{l}}$	$d_s$	$\zeta$	$\Lambda$	$\delta$	$〈D_i〉$	$D_{tW}$
25	1	1	1.8614	1.9746	1.951	0.0497	0.470	4/5	1.9723	1.892
9	1	1	1.6309	1.8928	1.806	0.2031	0.384	2/3	1.877	1.771
16	4	1	1.5	1.7935	1.659	0.3684	0.303	1/2	1.75	1.792
25	9	1	1.4307	1.7227	1.572	0.4683	0.259	2/5	1.6584	1.829
36	16	1	1.3869	1.6719	1.516	0.5333	0.232	1/3	1.5912	1.860

.

There are two kinds of generalization of the standard Sierpiński carpets in three dimensions—the Sierpiński cubes and the Menger sponges (see Refs. [141,142,143,144,145,146,147,148,149,150] and references therein). The peculiar properties of standard Sierpiński carpets and cubes and Menger sponges remain an active topic of research (see Refs. [151,152,153,154,155,156,157,158,159,160] and references therein).

3.4. The Space-Filling Curves

One of the most surprising mathematical discoveries of the nineteenth century was the existence of space-filling curves. The first plane-filling curve was constructed by Peano [10], who found a continuous and surjective map from an interval onto a square. Peano’s work was motivated by Cantor’s proof that the unit interval and the unit square have the same cardinality. The Peano curve may be constructed starting from a line segment and substituting it with the basic shape, as, for instance, shown in Figure 8a. After that, every one of the nine line segments is taken and substituted with the basic shape again. The square-filling Peano curve emerges in the limit of iterations through the sequences of square centers. It passes each point in a unit square, and it is everywhere continuous but nowhere differentiable [161,162,163,164]. However, the iterative construction does not assure a continuous one-to-one correspondence between the unit interval and the unit square. In fact, such a correspondence does not exist because any square-filling curve has multiple points due to self-intersections. Two notable variants of space-filling curves known as the Hilbert curve and the Sierpinski square snowflake are shown in Figure 8b,c, respectively. It is pertinent to point out that the space-filling curve always appears as a limit of the curve sequences but none as the curves in the sequence.

In this regard, it should be realized that a two-dimensional unit square cannot be homeomorphic to a one-dimensional unit interval because the square has no cutpoint, whereas all points of the curve (except the endpoints) are cutpoints. Therefore, a self-avoiding curve cannot fill the unit square. Although approximations of a space-filling curve can be self-avoiding as illustrated in Figure 7, the limits of the curve sequences have multiple points due to self-contact without self-crossing. Though, a space-filling curve can be everywhere self-crossing if its approximation curves are self-crossing. Space-filling curves can be also constructed in higher-dimensional spaces [165,166,167,168,169].

Space-filling curves are nowhere rectifiable, and so they form a special class of fractal curves [169,170,171,172]. It is noteworthy that space-filling curves do not obey the constitutive inequality (6) because the topological and fractal (e.g., box-counting) dimensions of the curve filling the n-dimensional space both are equal to the space dimension $n\ge 2$. Furthermore, taking into account that, generally, $d\le D_{tF}\le D$ [53], it is straightforward to understand that the topological fractal dimension of a space-filling curve is also equal to the space dimension. Due to multiple points, the connectivity dimension of the space-filling curve is equal to the fractal dimension, and so the spectral dimension is also equal to the space dimension because, generally, $d\le d_s\le d_{\mathcal{l}}$. Thus, the space-filling fractal curves obey the following equalities:

$$d_{\mathcal{l}}=D=D_{tF}=d_s=d=n,$$

(48)

which also are characteristic for the n-dimensional Euclidean patterns. A challenging question is whether there exists a dimension number which allows to distinguish between the space-filling fractal curves and Euclidean patterns.

3.5. Cantor Set and Totally Disconnected Fractals

The notion of connectedness plays a fundamental role in modern topology. The first definition of connectedness was suggested by Cantor in order to properly comprehend the notion of a continuum [173]. His definition appeals to the concept of distance. Cantor [7] states that a point-set $P$ is connected if for every two of its points $p,p\prime$ and an arbitrary number $ϵ>0$, there always exists a finite number of points $p_i\in P$, $i=\mathrm{1,2},\dots ,n$ such that the distances $p{p}_1, p_1{p}_2,\dots p_{n}p\prime$ are smaller than $ϵ$. A modern definition of the term was given by Arthur Moritz Schoenflies [174] without appealing to the notion of distance. The Schoenflies definition reads: a perfect set $T$ is called connected if it cannot be decomposed into subsets, each of which is perfect. Recall that after Cantor [7], a closed set is termed perfect if it does not contain isolated points.

After introducing the concept of a perfect set, Cantor explored the set of real numbers of the form $=c_1/3+\cdots +c_k/3^k+\cdots$, where $c_k$ is 0 or 2 for each integer $k$. This representation allows for a one-to-one mapping between the set of real numbers and the closed unit interval. Cantor [7] noted that $C=\left\{x={}_{3}0.c_1{c}_2\dots c_i\in \left\{\mathrm{0,2}\right\}\right\}$ is the infinite, perfect set, which is not everywhere dense in any interval, regardless of the interval length. Thus emerges the famous ternary Cantor set (see the history survey in Ref. [175]). The ternary Cantor set is compact and has the Lebesgue measure zero. Recall that a set is called compact if it is closed and bounded, such that every open covering of the set contains a finite subcovering. It was ascertained that every compact, perfect, and totally disconnected set is homeomorphic to the ternary Cantor set [176]. A necessary and sufficient condition for the existence of a Cantor set contained in a subset of the real line was established in Ref. [177]. Later, it was demonstrated that every real number between zero and one can be presented as a product of three elements of the ternary Cantor set [178]. In this regard, it was recognized that that the Cantor set is unique dense in itself compact metric space having only finite number of topologically distinct open subsets [179]. Due to these remarkable properties, the Cantor set plays a fundamental role in many branches of mathematics. Its special place in the theory of compact spaces is coined by the Hausdorff–Alexandroff theorem which states that any compact metric space is a continuous image of the Cantor set [180,181,182]. This theorem is of fundamental importance, not only for the theory of compact spaces but also in many other branches of mathematics, such as topology, geometry, probability theory, functional analysis, number and measure theories, etc. (see, for example, Refs. [182,183,184,185,186,187,188,189,190,191,192,193] and references therein).

Since the celebrated Cantor paper [7], it was realized that there are many different ways to construct totally disconnected (Cantor-like) sets. Geometrically, the ternary Cantor set can be constructed by repeatedly deleting the open middle thirds from a set of line segments, starting from the unit interval [0, 1], ad infinitum (see Figure 1a). It is easy to verify that the resulting set of points has a similarity dimension equal to $D_S=\ln 2/\ln 3$, while its Lebesgue measure is zero. If instead of the middle-third, one takes out the middle-$ϵ$ of each interval, the measure of the resulting (middle-$ϵ$) Cantor set is still zero and the fractal dimension is equal to

$$D=\ln 2/\left[\ln 2-\ln \left(1-ϵ\right)\right],$$

(49)

where $0<ϵ<1$, such that $0<D<1$ [191]. The average distance between points of the middle-$ϵ$ Cantor set is found to be equal to

$$\mathcal{L}=\left(1+ϵ\right)/\left(3+ϵ\right),$$

(50)

such that for the middle-third Cantor set, $\mathcal{L}=2/5$ [194]. Nonetheless, any middle-$ϵ$ Cantor set is homeomorphic to the middle-third Cantor set [195]. The chaotic attractors of many one-dimensional maps, such as the logistic maps, turn out to be topologically equivalent to the middle-third Cantor set [196].

Another generalization of the Cantor set is known as the Smith–Volterra–Cantor sets [197]. These sets are created by recursively removing the intervals of length ${\sigma }^k$ ($\sigma <1/3$) in the step $k (k=\mathrm{1,2},3,\dots )$ from each interval retained after the previous step, starting from the unit interval, ad infinitum. A Smith–Volterra–Cantor set has a finite Lebesgue measure which approaches one as $\sigma \to 0$. Accordingly, the Smith–Volterra–Cantor sets form part of the so-called fat Cantor sets [190]. The average distance between the points of the Smith–Volterra–Cantor set is equal to $\mathcal{L}=2/3\left(2-\sigma \right)$ [194]. Although Smith–Volterra–Cantor sets have a fractal (box-counting) dimension exactly equal to one, they are still homeomorphic to the middle-third Cantor set.

The Cantor set in a higher dimension (commonly referred to as the Cantor dust) can be constructed either by recursive iterations starting from unit cube ${\left[\mathrm{0,1}\right]}^n$, as shown in Figure 1b, or as the Cartesian product of $n$ orthogonal Cantor sets [198,199,200,201]. Like the Cantor sets, the Cantor dusts are totally disconnected uncountable sets of points with zero measure. Accordingly, the Hausdorff dimension of the Cantor dust is equal to the sum of the fractal dimensions of the corresponding Cantor sets, and so it can vary in the range of $0<D<n-1$, though all Cantor dusts are homeomorphic to the middle-third Cantor set, while their topological dimension is zero.

Thus, the totally disconnected fractals obey the following relations between dimension numbers:

$$D>D_{tF}=d=0,$$

(51)

whereas the definitions of the connectivity and spectral dimensions, as well as the fractal dimension of the random walk, are not straightforward. In fact, the definition (4) is applicable only to the path-connected fractals. Furthermore, in order to define a continuous random walk on the totally disconnected Cantor set (dust), one needs to relax the continuum requirement (see, for instance, Refs. [202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225]). Accordingly, the connectivity and spectral dimensions of the totally disconnected fractal are dependent on how the continuum requisite is handled (see, for review, Ref. [226] and references therein).

In particular, Lobus [205] constructed strong Markov processes on the Cantor set from Wiener processes by means of time change, killing, and space transformation. Consequently, the Brownian motion on the Cantor set is the trace of a suitable diffusion on the interval $\left(\mathrm{0,1}\right)$. Freiberg and Zahle [227] proposed to admit the jumps between the endpoints of the cutouts in the totally disconnected Cantor set. This allows to define the Markovian Lévy flights on the totally disconnected fractals which guide to a superdiffusion [226]. Conversely, Bakhtin [228] introduced the Lévy walks with a constant velocity on the totally disconnected Cantor set. These walks are symmetric, self-similar Markov processes leading to a subdiffusion on the totally disconnected fractals [226]. On the other hand, Takahashi and Tamura [210] considered the random processes defined as limits of suitably scaled random walks on the totally disconnected fractals, which lead to subdiffusion, as well as to superdiffusion [226].

In this regard, we noted that the totally disconnected fractal can be endowed with a suitable metric [229,230,231,232,233,234]. If this metric satisfies a stronger form of the triangle inequality

$$\mathcal{l}(x,y)\le \max \left\{\mathcal{l}\left(x,z\right),\mathcal{l}(z,y\right\}\forall x,y,z\in C,$$

(52)

then it is called an ultrametric or non-Archimedean metric [233]. A metric of the Cantor set is called regular if the topology induced by this metric is equivalent to the usual topology. A Cantor set with a regular ultrametric determines a weighted, rooted tree graph whose boundary is isometrically equivalent to that set [185,186,187]. In this context, it was proved that the ultrametric Cantor dust with the fractal dimension $D$ is bi-Lipschitz embeddable in $E^{\left[D\right]+1}$, where $\left[D\right]$ denotes the integer part of $D$ [234]. Consequently, the number of effective spatial degrees of freedom of a random walker on the ultrametric Cantor set is, obviously, $n_{\gamma }\le \left[D\right]+1$. Specifically, it was argued (see Ref. [226]) that the number of effective spatial degrees of freedom in the ultrametric Cantor dust embedded in $E^n$ is equal to

$$n_{\gamma }=n=\left[D\right]+1,$$

(53)

while the fractal dimension is in the range of $0<D<n$.

On the other hand, the instantaneous jumps between the endpoints of the cutouts admitted in the Freiberg and Zahle model [227] imply that the fractal dimension of the minimum path on the totally disconnected fractal is equal to

$$d_{\min }=D/n$$

(54)

and so the connectivity dimension of the totally disconnected fractal embedded in $E^n$ is equal to the embedding dimension, that is

$$d_{\mathcal{l}}=n=\left[D\right]+1>D,$$

(55)

such that relations (20) and (53) imply that the spectral dimension is equal to

$$d_s=d_{\mathcal{l}}=n>D,$$

(56)

while the fractal dimension of the random walk (Markovian Lévy flights) is

$${D_W=2d}_{\min }=2D/n<2,$$

(57)

and so the Markovian Lévy flights on the totally disconnected fractals lead to superdiffusion [226].

In the case of the Lévy walk with a constant velocity on the totally disconnected fractal, the fractal dimension of the minimum path is equal to $d_{\min }=1$. Accordingly, the connectivity dimension is equal to the fractal dimension, that is

$$d_{\mathcal{l}}=D<n=\left[D\right]+1,$$

(58)

such that the spectral dimension is equal to

$$d_s=2D-n<D,$$

(59)

and so the fractal dimension of Lévy walk on the totally disconnected fractal is

$$D_W=2D/\left(2D-n\right)>2,$$

(60)

while $D>n/2$ [226]. However, if the Markovian random process is defined as the limits of the scaled random walks on the totally disconnected self-similar set [210], then the fractal dimension of the random walk is equal to

$$D_W=D+1>n=\left[D\right]+1,$$

(61)

and so

$$d_s=2D/(D+1),$$

(62)

whereas the equalities in Equation (53) do not necessarily hold. Notice also that in this case, the relation in (61) differs from the relation (25) for the loopless fractals.

4. Conclusions

In this review, we pointed out that the constitutive inequality between fractal and topological dimensions can have either a geometric or topological origin, or both. Thereon, our issue is focused on the topological attributes of fractals. Hence, we first scrutinized the definitions of the dimension numbers characterizing the topological properties. Hereinafter, we argue that the main topological features of a fractal are associated with its connectedness, connectivity, ramification, and loopiness. Consequently, we suggest that these features can be specified by six basic dimension numbers which are generally independent from each other. However, this number may be reduced due to the relations between the basic dimension numbers, which are different for different kinds of fractals. Accordingly, we survey the paradigmatic fractals from a topological perspective. In this way, some challenging points are outlined.

In particular, we note that the long-range interactions between the points on a self-avoiding fractal curve can change its topological attributes. In this work, we reveal how the power law interactions affect the connectivity and spectral dimensions. We also state that the topological properties of the Sierpiński gasket shape obtained as the limit of the self-avoiding arrowhead curve differ from those obtained as the limit of the finitely ramified Sierpiński network. Thereon, we argue the need for additional dimension numbers in order to properly characterize the topological features of infinitely ramified Sierpiński carpets. The peculiar topological features of space-filling fractals are also highlighted. Finally, we discuss how to define the random walk and dimension numbers for totally disconnected fractals. We expect that our review will stimulate further studies on the topological features of deterministic and real-world fractals.