On the Topological Convergence of Multi-Rule Sequences of Sets and Fractal Patterns

Soft Computing https://doi.org/10.1007/s00500-020-05358-w

FOCUS

On the topological convergence of multi-rule sequences of sets and fractal patterns

Fabio Caldarola1 · Mario Maiolo2

Abstract In many cases occurring in the real world and studied in science and engineering, non-homogeneous fractal forms often emerge with striking characteristics of cyclicity or periodicity. The authors, for example, have repeatedly traced these characteristics in hydrological basins, hydraulic networks, water demand, and various datasets. But, unfortunately, today we do not yet have well-developed and at the same time simple-to-use mathematical models that allow, above all scientists and engineers, to interpret these phenomena. An interesting idea was ﬁrstly proposed by Sergeyev in 2007 under the name of “blinking fractals.” In this paper we investigate from a pure geometric point of view the fractal properties, with their computational aspects, of two main examples generated by a system of multiple rules and which are enlightening for the theme. Strengthened by them, we then propose an address for an easy formalization of the concept of blinking fractal and we discuss some possible applications and future work.

Keywords Fractal geometry · Hausdorff distance · Topological compactness · Convergence of sets · Möbius function · Mathematical models · Blinking fractals

1 Introduction ihara 1994; Mandelbrot 1982 and the references therein). Very interesting further links and applications are also those The word “fractal” was coined by B. Mandelbrot in 1975, between fractals, space-filling curves and number theory but they are known at least from the end of the previous (see, for instance, Caldarola 2018a; Edgar 2008; Falconer century (Cantor, von Koch, Sierpiński, Fatou, Hausdorff, 2014; Lapidus and van Frankenhuysen 2000), or fractals Lévy, etc.). However, it is only in the last few decades that and hydrology/hydraulic engineering as we will recall better fractals have known a wide and transversal diffusion and below. the interest of the scientific world towards them has seen The main characteristic of a fractal, as it is well known, is an exponential growth. In fact, fractals have been applied the property of self-similarity at different scales, and many in many fields, from the dynamics of chaos to computer abstract mathematical models have been created by focusing science, from signal theory to geology and biology, etc. on this property. In most cases, a fractal is in fact mathemati- (see for example Barnsley 1993, 2006; Bertacchini et al. cally described by a generating rule or an iterated mechanism, 2018, 2016; Briggs 1992; Falconer 2014; Hastings and Sug- but in the real world it is not difficult to find examples in which it clearly emerges that a single simple rule is not enough to Communicated by Yaroslav D. Sergeyev. build the fractal. So in recent years, there have been several attempts and studies, some very successful, to implement tra- B Mario Maiolo [email protected] ditional fractal theory. For example, a multifractal system is a generalization of a classical fractal which uses a continu- Fabio Caldarola [email protected] ous spectrum of exponents to describe its dynamics, in the place of a single exponent, given by the fractal dimension, 1 Department of Mathematics and Computer Science, in traditional models (see Bernardara et al. 2007; Falconer Università della Calabria, Cubo 31/A, 87036 Arcavacata di 2014;Harte2001). For many natural phenomena, it is in fact Rende (CS), Italy completely insufficient to use a model that provides a single 2 Department of Environmental Engineering, Università della fractal dimension; multifractal systems have been typically Calabria, Cubo 42/B, 87036 Arcavacata di Rende (CS), Italy 123 F. Caldarola, M. Maiolo applied in contexts with different mass concentrations and in needed. This could be the case of “blinking fractals” or some- chaotic dynamics, for instance, to the Sun’s magnetic field, thing similar to them. human heartbeat and brain activity, turbulent dynamics in The idea of blinking fractal was introduced by Y.Sergeyev fluids, meteorology, geophysics, but also finance, internet in 2007 (see Sergeyev 2007) to describe fractals that assume traffic, and others (see Falconer 2014;Harte2001; Ivanov different shapes or configurations during their development, et al. 1999; Stanley and Meakin 1988; Veneziano and Essiam with a cyclical order. As far as the knowledge of the authors, 2003 and their references). they were successively applied only in Sergeyev (2011)up Another interesting example is given by the superfrac- until now, to model a process of growth in biological systems tal formalism introduced in 2002 by Barnsley, Hutchinson (like the growing of trees through the different seasons of the and Stenflo. The class of deterministic fractals is not too year).1 We also inform the reader that in Sergeyev (2007) and rich to study effectively the real world, because nature often (Sergeyev 2011) blinking fractals are studied with the support mixes deterministic aspects and casuality in its patterns. of a new computational methodology developed by Sergeyev Then, superfractals are precisely the models that are halfway himself in the early 2000s to write and to perform calculations between deterministic and completely random fractals, and with infinite and infinitesimal numbers in a handy and very present characteristics of both groups (for a comprehensive easy way as in the ordinary sets of natural or rational num- introduction to 1- and V -variable superfractals, see Barnsley bers N and Q (see Sect. 5 for some more information and the 2006). references Amodio et al. 2017; Antoniotti et al. 2020a, b;Cal- In their experience in hydraulic contexts, the authors have darola 2018b; Caldarola et al. 2020b; Sergeyev 2013, 2008, often found semblances and fractal properties emerging in the 2009, 2010, 2016, 2017). More precisely, a blinking frac- course of various researches, especially in those conducted tal is specified in Sergeyev (2007) as an “object constructed by the second author since the 90s. For example, the applica- using the principle of self-similarity with a cyclic application tion of fractal models to hydrological basins, natural channel of several fractal rules.” networks, but also to urban rainfall catchments, marine waves The main purpose of this paper is to make a topological- actions, shallow waters, and much more, has become a com- geometric contribution for the development and a concrete mon topic in the scientific literature (see, e.g., Bernardara future use of fractal models which present periodic changes et al. 2007; Rodríguez-Iturbe and Rinaldo 2001; Sivakumar or alterations, as in the case of blinking fractals. In particular, 2017; Veltri et al. 1996; Yang et al. 2014). Recently, more- after some preliminary material recalled in Sect. 2, we exam- over, the theory of fractals and fractal dimension has started ine in Sect. 3 the second example (and the first geometrically to be applied also to artificial infrastructures such as water constructed) given in the introduction of the article (Sergeyev distribution networks, in particular in urban agglomerations, 2007), the one where the name blinking fractal is used for the as in Di Nardo et al. (2017); Diao et al. (2017); Kowalski first time. We deeply investigate Sergeyev’s sequence Sn n et al. (2014); Qi et al. (2014); Wu et al. (2009). The same as described in Sergeyev (2007 ), firstly by considering sep- authors are conducting some research using tools such as arately the two subsequences An n and Bn n made up by algebraic graph theory, fractal geometry and a new system the odd and even indexed element, respectively. The sequence S of local indices (see Bonora et al. 2020a, b, c; Caldarola and Bn n clearly converges to a fractal equal to the intersec- Maiolo 2019, 2020a, b) in the study of water networks: in tion of all Bn, instead, such a property does not hold for the second case, the biggest challenge is to find a determin- An n. Hence we begin a series of computations that allow istic fractal model, without random components, but which to find the precise Hausdorff distance between any two ele- takes into account some temporal periodicities, for example, ments of An n (Proposition 2) and Bn n (Proposition 3), on a daily, weekly, seasonal basis, which characterize urban to get successively a general formula that gives the distance water networks. The same challenge arises in the study of the for any two elements of Sn n(Theorem 1). As elementary S fractal aspects of water demand, where a superfractal or mul- consequence, the sequence Sn n itself converges to which tifractal model does not seem to be the most effective or the has Hausdorff dimension 3/2. most appropriate for the purpose, as well. Furthermore, these In Sect. 4 we use a three-rules system to define a sequence models do not best respond to the request for structural sim- of plane shapes Xn n that has two subsequences Yn n and plicity and ease of use by technicians and engineers who work Cn n that converge to two different fractals, Y and C respec- on real networks and who struggle constantly with important tively. Proposition 4 computes their exact distance. complications and difficulties from a computational point of Section 5 addresses conclusions and, from the point of view. view of traditional mathematics, proposes to define a blinking For the reasons exposed above, the authors are very inter- fractal of order m simply as a m-tuple of traditional fractals ested to investigate and develop a fractal model as simple as possible, but that can easily involve periodic changes as 1 The reader can also see the book (Kaandorp 1994) for a rich com- pendium of fractal models applied to growth processes in biology. 123 On the topological convergence of multi-rule sequences of sets and fractal patterns

N (i.e., of blinking fractals of order 1). If we adopt this deﬁni- If A and B are two nonempty subsets of R we deﬁne ( , ) tion, Sergeyev’s sequence Sn n generates a blinking fractal their Hausdorff distance dH A B by of order 1 and Xn n a blinking fractal of order 2. dH (A, B) := inf ε ≥ 0 : A ⊆ B{ε} and B ⊆ A{ε} . (1)

2 Preliminary definitions and results It is simple to observe that, equivalently, we can also define dH (A, B) as This section collects and explains some necessary notations and definitions together with some fundamental, well-known dH (A, B) = max sup inf d(a, b), sup inf d(a, b) . ∈ ∈ properties of the recalled objects. a∈A b B b∈B a A First of all, the symbols Z, N and N+ denote, as usual, the sets of integers, of nonnegative integers and positive inte- The function dH is not a metric in general, but it becomes so gers, respectively. The standard symbol R stays for the set of if we consider the restriction of dH to the family + real numbers and R for the set of nonnegative real ones. 0 N N Given two integers n ≤ m, the writing [n .. m] is the H R := K ⊂ R : K compact and K =∅ . most common notation for discrete intervals, i.e., we set [n .. m]:=[n, m]∩Z where [n, m] stands for the real H RN is usually referred to as the hyperspace of nonempty interval as usual. A sequence will be indicated as a , N n n∈N compact subsets ofR , and it is not difficult to see that, RN , H RN , an n, or sometimes simply as an. similarly to d , the pair dH is a complete We now continue by briefly recalling and explaining metric space. Moreover, if A, B ∈ H RN , then it is easy to some basic notations in fractal geometry that will be fre- show that quently used in the next sections. For the benefit of the non-mathematical reader interested in blinking fractals, we dH (A, B) ∈ ε ≥ 0 : A ⊆ B{ε} and B ⊆ A{ε} , (2) will use, both here and in the next sections, a language as simple and elementary as possible, also providing from time i.e., the infimum in (1) is actually a minimum. to time enough details to facilitate reading. Lastly, we need to recall here a basic result from algebraic ∈ N+ For any N , we indicate by d the standard Euclidean combinatorics on words, because when we will draw conclu- RN ∈ RN ∈ R+ [ ] metric on .Forx and r 0 , Br x denotes sions in Sect. 5 we will have too few space. Classic references RN the closed ball in with radius r and center x, while, if in the field are the first two Lothaire’s books (Lothaire 1983) > ( ) ε r 0, Br x denotes the correspondent open one. If is a and (Lothaire 2002). RN nonnegative real number and A is any subset of ,letA{ε} Let A be an alphabet, i.e., a set of some distinct symbols ε be the -hull of A, that is, called letters;aword w over A is a finite sequence of elements written N A{ε} := x ∈ R : d(x, a) ≤ ε for some a ∈ A w = a1a2 ...an = Bε[a] . ∈ + a A with n ∈ N and ai ∈ A. We also say that w has length n, and we write |w|=n.ThesetA+ of all words over A, i.e. Remark 1 In literature A{ε} has many names as, for instance, ε ε ε + + the -fattening,the -parallel body, -dilatation,oralsothe A := w = a1a2 ...an : n ∈ N and ai ∈ A , ε-neighborhood of A. The last name is more common but it N is preferable for the set Nε(A) := x ∈ R : d(x, a)< is a semigroup with the operation of concatenation, and if ε for some a ∈ A , because we cannot speak of open and 1 denotes the empty word then A∗ := A+ ∪{1} is a free ε R2 N ( , ) closed -neighborhoods. In , for example, 1 B1 0 0 monoid (called the free monoid over A) because 1 acts as and B (0, 0) are both open sets, equal to B (0, 0) , w ∈ ∗ 1 {1} 2 neutral element. A word A is said to be primitive if N [( , )] = ( , ) = [( , )] = w = w = n instead 1 B1 0 0 B2 0 0 B1 0 0 { } it is not power of another word, i.e., if 1 and u 1 ∗ B2[(0, 0)]. for some u ∈ A and n ∈ N implies u = w and n = 1. It is important also notice for the reader that, for our pur- The following result is a consequence of the so-called defect poses, it will be quite more convenient to work with enlarged theorem (see in particular Lothaire 1983, Proposition 1.3.1) sets of the kind A{ε} rather than Nε(A), hence in (1)wegive or can be also viewed as an immediate corollary of Fine the definition of Hausdorff distance accordingly, among the and Wilf’s theorem (see Lothaire 1983, Proposition 1.3.5 or many possible ones. Lothaire 2002, Proposition 1.2.1). 123 F. Caldarola, M. Maiolo

Proposition 1 If xn = ym for some x, y ∈ A∗ and n, m ∈ N, then x and y are both powers of some z ∈ A∗. In particular, for each word x ∈ A+ there exists a unique primitive word w ∈ A+ such that x is a power of w.

Let us make a few comments on the last claim in the previous proposition. If x is a nonempty word, it is rather obvious + that there exists a primitive word w ∈ A such that x is (a) S0 (b) S1 a power of w (use, for example, a trivial induction argu- ment on the length of x). Instead it is less obvious that if x = wn = vm, with w and v primitive words, then w = v: It follows from the ﬁrst part of the same proposition.

3 Example: a deeper study on a “two-rule fractal” (c) S2 (d) S3

As first example, we want to consider a sequence presented in the initiator paper (Sergeyev 2007) to blinking fractals: Here we will deepen its analysis from a geometrical point of view also making various computations that allow to prove some sharp results. From now on, we will often use the symbol I to denote the closed interval [0, 1]⊂R. We go to define a sequence R2 (e) S4 (f) S5 of compact subsets Sn n∈N contained in and, to have a clearer view of the constructive process, it is convenient to imagine the elements Sn with an even index n colored in blue and those with an odd index in red. The initial element S0 is the square I × I ⊂ R2, colored hence in blue, and the element S1 is given by 2 2 S1 := (x, y) ∈ R :|x − 1/2|+|y − 1/2|≥1/2 ∩ I , (g) S6 (h) S7 that is, the red area consisting of the four isosceles right Fig. 1 The first eight elements of the sequence Sn n, starting from S0, triangles with side 1/2 positioned in the four corners of I 2,as the square of side 1. In the left column, the elements of even index, that shown in Fig. 1b. To describe S1 by words, we could also say is blue, and in the right column those of odd index, red that it has a vague shape of a square-rhomboid frame (square externally and rhombus internally). Note, finally, that in the 3t passage from the blue S to the red shape S we lose half of S2t+2, just replace each of the 2 small red frames that make 0 1 t the area. up S2t+1 with a blue shape like S2,but4 times smaller. To consider in future more complex model, we want to Consider now S2: It consists of 8 small squares of side 1/4 out of a total of 16 in which I 2 is divided, as illustrated remark as our sequence Sn n can also be viewed as the result in Fig. 1c. In particular, there are four squares in the center of the successive application of two rules: R1 transforms a blue square of side l into a red frame like S1 with the same and one in each corner. Note that in the passage from S1 to external side l, and R2 transforms a red frame F of side l into S2 we have no loss of area: All the red area of S1 is in fact a blue shape like S2, with the same area and diameter of F transformed in the equivalent blue area of S2. (see Fig. 2). The process to obtain recursively the subsequent elements For convenience in our presentation and to consider sepa- of Sn should be clear from Fig. 1.Atthestep2t, t ∈ N, n rately the subsequences of S with odd and even indices, the unitary square I 2 is divided into 42t small squares of side n n t 3t we set, for every n ∈ N, (1/4) each one; 2 of them are blue and form S2t . Then, to obtain S2t+1, we only have to replace each blue square with a small red frame as in the transition from S0 to S1. And to get An := S2n+1 and Bn := S2n. (3) 123 On the topological convergence of multi-rule sequences of sets and fractal patterns

(a)

(a) (b) n Fig. 3 The second and third element of the sequence i=0 Ai n∈N for an immediate comparison with An n∈N (b)

Fig. 2 The transformation rules R1 and R2. Using them, S2t+1 is , > ≥ 2t Proposition 2 For all integers n m with m n 0 we have obtained from the application of the rule R1 for each of the 4 blue −t squares of side 4 constituting S2t ,andS2t+2 is given by applying the √ 2t rule R2 at each of the 4 red frames that make up S2t+1 2 d A , A = . H n m n+1 (7) 4 The sequences A and B have different or common n n n n Proof = ≥ characteristics on dependence of the point of view, as we will Consider first the case n 0, and let m 1beany fixed integer. Since A ⊂ I 2 ⊆ A √ and see in the following. The simpler one is the second: It is in m 0 2/4 fact a nested sequence of closed subsets of I 2, that is, √ ⊃ ⊃ ⊃ ...⊃ ⊃ ⊃ ... 1 1 2 B0 B1 B2 Bn Bn+1 (4) , ∈ Am − A for all ε ∈ 0, , 2 2 0 {ε} 4 Since each Bn is nonempty, the family Bn : n ∈ N has then the finite intersection property (FIP) and, as well-known in general topology, the intersection of all its elements B n∈N n √ H R2 is nonempty as well and belongs to . In this case, it is ε ≥ : ⊆ = 2, +∞ . 0 Am A0 {ε} (8) moreover obvious that the sequence Bn n itself converges 4 to the mentioned intersection which will be denoted by S;in symbols Let now Q be the subset of I 2 formed by the 12 points as below lim Bn = Bn =: S . (5) n→∞ n∈N 1 3 Q = , 0 , , 0 , 4 4 It is very easy to visualize Sergeyev’s fractal S through the S 1 1 1 1 decreasing sequence (4): has a shape that presents some 0, , , , 1, , 4 2 4 4 vague similarities with the well-known Vicsek cross fractal and the Sierpiński carpet. 1 1 3 1 , , , , (9) The former sequence An n∈N is even more interesting 4 2 4 2 than B both because it seems unnoticed in the litera- n n∈N 3 1 3 3 ture, and it is less evident that it yields a fractal; for instance, 0, , , , 1, , 4 2 4 4 on the contrary of what observed in (4), this time there are 1 3 no inclusion relations in the sense that , 1 , , 1 ; 4 4 Ai ⊆ A j for somei, j ∈ N ⇒ i = j , (6) ⊂ 2 ⊂ ⊂ ⊂ since A0 I Q{1/4} and Q Am, then A0 ⊂ ( / , ) ∈ and the intersection An has little to do with the limit of Q{1/4} Am { / }, but note that we also have 1 2 0 ∈N 1 4 n − ε ∈[ , / [ A0 Am {ε} for all 0 1 4 . Therefore this means the sequence An n∈N, if it exists (see, for example, Fig. 3). Recalling the definition of Hausdorff distance given in (1), we can state the following 1 ε ≥ 0 : A ⊆ Am = , +∞ (10) 0 {ε} 4 123 F. Caldarola, M. Maiolo √ and, recalling (1) and (8), we conclude that whenever ε< 2 4n+1. This yields A A for √ m n {ε} √ all ε< 2 4n+1, and recalling (12) and (13) we finally √ 2 1 , = n+1 > ≥ dH A0, Am = inf , +∞ ∩ , +∞ conclude that dH An Am 2 4 for all m n 1. √ 4 4 (11) = 2, A consequence of the previous proposition is that An n∈N 4 2 is a Cauchy’s sequence in the complete hyperspace H R , + 2 for every m ∈ N . and hence, it converges to some S ∈ H R . The use of = Now consider the complementary case of n 0just Proposition 2 is not the shortest way to show that An n examined: let m > n ≥ 1 be fixed integers and recall that converges, but it has several advantages such as that of pro- = 3n An S2n+1 is made up by 2 red frames of (external) side viding the reader with an effective and easy tool to investigate −n 3n length 4 .IfFi , i ∈ 1 .. 2 , is one of these frames the dynamics of other fractal processes. Moreover, Proposi- then, in the transition from An to Am, Fi is replaced by a red tion 2 allows to treat the sequence An n independently from n shape Gi equal to Am−n but 4 times smaller. From the case Bn n and will be a piece of the proof of Theorem 1 just as n = 0 discussed above, and in particular from (8) and (10), the next proposition will constitute another. As regards the we immediately obtain fractal S , it is actually equal to S and it is quick to prove √ directly. We instead prefer to wait and to see it as corollary ε ≥ : ⊆ = 1 · 2, +∞ of Theorem 1. 0 Gi Fi {ε} 4n 4 The next proposition establishes a twin formula of (7)for √ the nested sequence Bn n. The proof uses the same pattern = 2 , +∞ and is easier than that of Proposition 2; in any case, we will + 4n 1 give some details for completeness. and Proposition 3 For all n , m ∈ N, n < m, we have ε ≥ : ⊆ = 1 · 1, +∞ 1 0 Fi Gi {ε} dH Bn, Bm = . (14) 4n 4 4n+1 = 1 , +∞ . Proof Let first n = 0 and m ≥ 1befixed.IfQ ⊂ I 2 is the set n+1 ⊂ = 4 defined in (9), we have Q Bm and, consequently, B0 I 2 ⊂ Q{ / } ⊂ B .But (1/2, 0) ∈ B − B for Since the ε-hull of a union of subsets is equal to the union of 1 4 m {1/4} 0 m {ε} all ε ∈[0, 1/4[, then we conclude that the ε-hulls, then 1 3n 3n 2 2 dH B0, Bm = . 4 A = G ⊆ F √ m i i 2/4n+1 = = > ≥ = i⎛1 ⎞i 1 Let now m n 1 be fixed. Recall that Bn S2n is 3n (12) 3n n 2 constituted by 2 squares of side length 1/4 and, if Di , i ∈ ⎝ ⎠ √ 3n = Fi = An + 1 ..2 , is one of them, then it is replaced, in the transition 2/4n 1 = √ i 1 2/4n+1 from Bn to Bm, by a blue shape Ei similar to Bm−n with n ratio 1/4 . Since D ⊂ E + ,asin(13) we obtain i i {1/4n 1} and similarly ⊂ Bn Bm {1/4n+1}, but observing that 23n 23n 1 , ∈ − A = F ⊆ G + 0 Bm Bn n i i {1/4n 1} 2 · 4n {ε} i=1 i=1 ⎛ ⎞ (13) 23n n+1 n+1 for every ε<1/4 , we finally get dH Bn, Bm = 1/4 = ⎝ ⎠ = . Gi Am {1/4n+1} for all m > n ≥ 1. = i 1 {1/4n+1} A general formula for the distance of two elements of the original Sergeyev’s sequence S is given by the follow- Now, considering the frame, say F1, inside the square n n∈N 0, 1/4n ×[0, 1/4n], note that its central point ing theorem. During its proof, we will see as (15) reduces, ≡ ≡ ( ) ≡ ≡ ( ) when n m 1 mod 2 and n m 0 mod 2 respec- 1 1 tively, to Formulas (7) and (14) stated for An n and Bn n, , ∈ Am − An 2 · 4n 2 · 4n {ε} resp. 123 On the topological convergence of multi-rule sequences of sets and fractal patterns ( ) ( )+ Theorem 1 For all integers m > n ≥ we have S ( ) t n · / = t n 1 0 2t m {ε} is 1 4 1 4 1 4 . On the other hand, ( ) the minimum ε to get S ( ) ⊆ S ( )+ is 1 4t n · √ n−6n/2−4+max{0, 1+n/2−m/2} √ √ 2t m 2t n 1 {ε} ( )+ dH Sn, Sm = 2 . (15) 2 4 = 2 4t n 1, hence √ Proof Let n = 2t(n) +r(n) and m = 2t(m) +r(m), where 1 2 ( ), ( ) ∈ N ( ), ( ) ∈{ , } dH S t(n)+ , S t(m) = max , t n t m and r n r m 0 1 . First we distinguish 2 1 2 4t(n)+1 4t(n)+1 ( ) ( ) four cases on dependence of r n and r m , and then draw √ (19) conclusions. 2 ( ) = ( ) = = ( ) = = . Case 1 If r n r m 0 then n 2t n and m 4t(n)+1 2t(m), and by Proposition 3 we have Conclusions. Now we just have to summarize the Formu- dH S2t(n), S2t(m) = dH Bt(n), Bt(m) las (16)–(19), arising from the four different cases, in a new one, and it can be made as follows 1 (16) = . t(n)+1 4 dH Sn, Sm = dH S2t(n)+r(n), S2t(m)+r(m) ( ) = ( ) = = ( ) + Case 2 If r n r m 1, i.e., n 2t n 1 and √ r(n) + max{0, 1 + t(n) − t(m)} m = 2t(m) + 1, by Proposition 2 we get 2 = 4t(n)+1 dH S2t(n)+1, S2t(m)+1 = dH At(n), At(m) √ √ n−2n/2+max{0, 1+n/2−m/2} (17) 2 = 2 . = 4t(n)+1 4n/2+1 √ n−6n/2−4+max{0, 1+n/2−m/2} Case 3 Let r(n) = 0 and r(m) = 1, i.e., n = 2t(n) = 2 , = ( ) + ⊃ and m 2t m 1. Since S2t(n) S2t(m)+1, we only have ε ⊆ to find the minimum such that S2t(n) S2t(m)+1 {ε}.For where x denotes the floor of a real number x. this purpose, we have to enlarge S2t(m)+1 in two directions: inward until the rhomboid holes of the 23t(m) red frames From Eq. (15), it follows immediately constituting S2t(m)+1 are closed, and outwards until S2t(n) is √ n − 6n/2−4 + 1 covered. The former minimum enlargement is given by d S , S ≤ 2 H n m √ √ √ n − 6(n − 1)/2 − 3 ≤ 2 = 2−n, , = 1 · 2 = 2 dH S2t(m)+1 S2t(m) ( ) ( )+ 4t m 4 4t m 1 hence Sn is a Cauchy sequence and converges to S and the second by n∈N defined in (5) because the subsequence S2n n∈N converges S 1 to . dH S2t(m), S2t(n) = 4t(n)+1 Lastly, let us now calculate the Hausdorff dimension of Sergeyev’s fractal S. Many well-known techniques can be (recall Proposition 3). Hence used to this purpose; for example, S can be very easily viewed √ as the attractor,orinvariant set,ofaniterated function system 2 1 (IFS) consisting of 8 similarities all with ratio 1/4. Since dH S2t(n), S2t(m)+1 = max , 4t(m)+1 4t(n)+1 it satisfies Moran’s open set condition, then the Hausdorff ⎧ √ dimension dimH , the box-counting dimension dimB, and the ⎪ (18) ⎨⎪ 2 ( ) = ( ) similarity dimension s are all equal and can be immediately ( )+ if t n t m 2 = 4t n 1 computed as follows ⎪ ⎩⎪ 1 if t(n)

for all i, j ∈[1 .. 9]. For any t ∈ N+, we continue in this way to introduce inductively the notation Σt , i1,i2,...,it where the superscript t means that they are square of t side length 1/3 and the subscripts i1, i2,...,it (or the t-tuple (i1, i2,...,it ) if one prefers) are the “coordi- (a) 3 nates” for the position of the square. Pt hence acts in the obvious way as seen in (20) and (21) for the special cases t = 1 and t = 2, respectively; we in fact deﬁne P Σt := Σt t i1,i2,...,it i1,i2,...,γ (it )

(b) + for all t ∈ N and i1, i2,...,it ∈[1 ..9]. Fig. 4 The transformation rules SC and SM . Both transform a square Weare now ready to describe P which acts on the squares 2 of area l to a shape, of different form, but with the same area equal to Σt at different levels likewise a “composition” of 2/ i1,i2,...,it 4l 9 + the Pi . Formally, for all t ∈ N and i1, i2,...,it ∈[1 .. 9],weset 4 A “three-rule fractal” P Σt := Σt . (22) i1,i2,...,it γ(i1),γ (i2),...,γ (it ) Now we describe a fractal based on the cyclic application of three rules, SC , SM and P, acting on a sequence Xn of n∈N We now deﬁne the sequence Xn n∈N by using alterna- geometric shapes in the real plane. The rules act as follows. tively the rules deﬁned in (i) and (ii), organized in a “4-cycle” as follows

(i) Consider any square Σ of side l in the plane. SC and S , P, S , P, S , P, S , P, etc. (23) SM subdivide it in 9 smaller squares of side l/3, then SC C M C M select the four of them in the corners of Σ, instead SM 2 takes the middle square for each side of Σ,asshownin We start by setting X0 := I , and then we continue recur- Fig. 4a, b, respectively. sively by using the sequence (23)asbelow (ii) To get P we need to deﬁne a family of actions {Pi : i ∈ + 2 N } as follows. Consider the unit square I as consist- X1 := SC (X0), Σ1,Σ1,...,Σ1 / := ( ) = ( ◦ )( ), ing of 9 smaller squares 1 2 9 of side 1 3 X2 P X1 P SC X0 Σ1 := ( ) = ( ◦ ◦ )( ), and disposed anticlockwise starting from 1 with cen- X3 SM X2 SM P SC X0 ( / , / ) Σ1 ( / , / ) := ( ) = ( ◦ ◦ ◦ )( ), (24) ter in 1 6 1 6 , 2 centered in 1 2 1 6 , and so on X4 P X3 P SM P SC X0 Σ1 ( / , / ) := ( ) = ( ◦ ◦ ◦ ◦ )( ), until 8 with center in 1 6 1 2 , and lastly the central X5 SC X4 SC P SM P SC X0 Σ1 Σ1 square 9 (see Fig. 5a). P1 acts on the i as the cycle etc. γ = ( , , , , , , , ) 1 2 3 4 5 6 7 8 belonging to the symmetric group Sym(9) acts on [1 ..9]. In other words, The ﬁrst few elements of the sequence Xn n are shown in Figs. 6, 7 and 8. ⎧ 1 Let Cn ∈N and Yn ∈N be the subsequences of Xn ∈N ⎨⎪Σ + if i ∈[1 ..7], n n n i 1 obtained by setting 1 1 1 P1 Σ := Σγ( ) = Σ if i = 8, (20) i i ⎩⎪ 1 Σ1 if i = 9. 9 Cn := Xn+2n/2 and Yn := Xn+2n/2+2 (25) Σ1 : ∈[ .. ] for all n ∈ N. Note that the sets Cn : n ∈ N and Yn : n ∈ Therefore, if A is a subcollection of i i 1 9 , N are disjoint and their union is the whole family X : n ∈ the meaning of P1(A) is clear. n ∈[ .. ] Σ1 For any i 1 9 consider now the square i as Σ2 ,Σ2 ,...,Σ2 3 consisting of 9 smaller squares i,1 i,2 i,9 of The attentive reader will think of using a pair of coordinates in base / 2 Σ2 Σ1 3: This has the obvious advantage of easily positioning the square in side 1 3 ; P2 acts on the i, j as P1 acts on the i , i.e., through the permutation γ ; more precisely we set question, but the permutations Pi would act in a less simple way to write. Since our aim is to explain in as clear and elementary a way as possible the dynamics of the permutations Pi whichplayanessential role in this section (as opposed to the exact position of the squares), Σ2 := Σ2 P2 i, j i,γ ( j) (21) then we have consequently chosen a convenient notation system. 123 On the topological convergence of multi-rule sequences of sets and fractal patterns

(a) (b) (a) (b)

(c)

Fig. 5 In a it is shown as the unit square I 2 is subdivided into 9 squares Σ1, ∈[ .. ] / (e) i i 1 9 , of side length 1 3. In b, the arrows represent the (f) action of P1. In Subﬁgure c, it is schematically illustrated the action of P2 Fig. 7 Continuing Fig.6, here are represented the successive six elements of the sequence Xn n∈N, that is, from X4 to X9

(a) (b)

Fig. 8 The element X9 is rather difficult to visualize in Fig. 7f, and it is increasingly difficult for the subsequent elements in the sequence Xn n∈N.BeingX9 made up by four copies of X8 placed in the corners 2 of I , the reader may find it helpful to represent X9 as in (a). Similarly, X10, X11 and X12 can be visualized through four copies of X7, X10 and X9 arranged as in (b)–(d), respectively, and so on from X13 onwards (c) (d) and Fig. 6 The first four elements of the sequence Xn n∈N starting from 2 X0 = I Y0 = X2, Y1 = X3, Y2 = X6, Y3 = X7, Y4 = X10, Y5 = X11, Y6 = X14, Y7 = X15, etc., N . As way of example, their first few elements are respectively.

Remark 2 C0 = X0, C1 = X1, C2 = X4, C3 = X5, It is immediate to recognize that the sequence = , = , = , = , , C4 X8 C5 X9 C6 X12 C7 X13 etc. Cn n is the standard one to construct the 2-dimensional 123 F. Caldarola, M. Maiolo 1 Cantor dust C which is the plane, i.e., 2-dimensional version x = (− )n/2 − , n ∈ N; n 1 2 + of the best knownCantor set contained in the interval [0, 1]. 1 n The subsequence Yn n, instead, converges to a fractal Y that has a “dust form” like C but is different and, actually, very both sn n∈N and xn n∈N present alternatively positive and far from it as the following proposition specifies. negative terms, but while the former converges to a limit √ = S s 0 likewise Sn n∈N converges to , the second has two Proposition 4 We have d (C, Y ) = 5 6. H main subsequences convergent respectively to 2 and−2 likewise X has two subsequences C and Y Proof n n∈N n n∈N n n∈N Considering the set convergent to C and Y , respectively. From the examples and all the discussion made from the 1 1 2 1 1 5 2 5 Q = , , , , , , , ⊂ Y previous sections until now, we therefore propose the follow- 3 6 3 6 3 6 3 6 ing definition.4 ⊂ √ ⊂ √ + we easily get C Q { 5/6} Y { 5/6}. But if we define Definition 1 A blinking fractal B of order m ∈ N is simply an m-tuple of (not necessarily distinct) traditional fractals 1 1 1 := ( , ) ∈ R2 : ≥ − ∧ ≥ ∨ ≥ W x y y x x y m 2 3 3 N B = (B1, B2,...,Bm) ∈ H R (27) we also have Y ⊂ W and, consequently, d (0, 0), Y ≥ √ such that the word B B ...B is primitive (recall the last d (0, 0), W = d (0, 0), (1/3, 1/6) = 5 6 (where the 1 2 m part of Sect. 2). distance d(a, B) is defined as usual to be inf{d(a, b) : b ∈ B} for all a ∈ R2 and B ⊆ R2). Therefore (0, 0) ∈ C − W{ε} ⊂ √ A traditional fractal can be viewed as a blinking fractal of C − Y{ε} for all ε< 5 6, and we obtain inf ε>0 : C ⊂ √ order 1, called also a simple fractal. For example, Sergeyev’s { } = Y {ε} 5 6. fractal S, analyzed in Sect. 3, is a blinking fractal of order 1, ⊂ On the other hand, for example,√ Y C{1/3} is trivial, then while the one presented in Sect. 4 is a blinking fractal B of ( , ) = we conclude that dH C Y 5 6 as we wanted. order 2 that can be written B = (C, Y ). It is important to note that in virtue of the unicity stated As regards the Hausdorff dimension it is trivial that in Proposition 1, the order m of a blinking fractal is well dim C = dim Y = ln 4/ ln 3 ≈ 1.26. H H defined. Example 1 Multi-rule fractals are very interesting to investi- Example 2 { , }⊂H R2 gate in less elementary or easy cases than the previous one. Consider the set of fractals C Y It is important to observe that the greater number of rules defined in the previous section. We can obviously form does not correspond to a more complex fractal in general: the reader, for example, can study the sequence Zn , – 2 blinking fractals of order 1 as well as 2 blinking fractals n∈N produced by the previous system by using only the first two of order 2 B = (C, Y ) is in fact distinct from B = 2 ( , ) of the three rules SC , P and SM , i.e., starting from Z0 = I Y C ; 4 − = and applying successively SC , P, SC , P, SC , P, and so –2 4 12 blinking fractals of order 4 in fact the on. writings (C, C, C, C), (Y , Y , Y , Y ), (C, Y , C,Y ) and (Y , C, Y , C) do not represent a blinking fractal ; –23 − 2 = 6 blinking fractals of order 3, 25 − 2 = 30 5 Conclusions and future work blinking fractals of order 5, etc. The sequences Sn n∈N and Xn n∈N, studied respectively See Proposition 5 below for a general formula that counts the in Sects. 3 and 4, could both be called “blinking sequences” number of blinking fractals. because they exhibit two alternating geometric shapes. To make a really skinny simplification, they behave similarly to The study of multi-rule and blinking fractals seems very the following two real sequences interesting both from a pure mathematical point of view and

n 1 4 sn = (−1) , n ∈ N, (26) First note that in this paper, following modern customs, we have 1 + n avoided to give a precise definition of (traditional) fractal; see the ubiq- uitous discussions on the issue present both in fractal monographs, but and also online. 123 On the topological convergence of multi-rule sequences of sets and fractal patterns for applications, and much work can be done in this direction positive and the other negative),5 while in this paper and in in our opinion. Just as a small example, the following ques- Weierstrass analysis it converges to zero, as we already noted. tions arise immediately considering the sequence Zn n∈N We end the paper by giving an enumerating formula for the defined in Example 2, and many other multi-rule sequences number νk(m) of blinking fractals of order m obtainable from can be evaluated and investigated. a basic set {F1, F2,...,Fk} of k distinct simple fractals. For this purpose we recall that in number theory the Möbius μ : N+ →{− , , } (i) Does the sequence Zn n∈N generate a blinking fractal? function 1 0 1 is defined by (ii) If yes, of which order? ⎧ ⎪ ⎪1ifn = 1, ⎨⎪ Also from an applied point of view, blinking fractals seem (−1)t if n is the product of t distinct rich of applications in many fields. Just to stay in the hydraulic μ(n) = ⎪ prime numbers, one, remember what was mentioned in the Introduction about ⎪ ⎩ , water demand in urban centers. In our experience, it is easy to 0 if n has some multiple prime factor find recurrent fractal-like patterns on different sizes and time or, equivalently, μ(n) can be defined as the sum of the prim- scales, also very small, in the water demands, and we think itive nth roots of unity (cf. the definition of primitive word that some rather jagged plottings are in part due to the speed in Sect. 2). with which modern taps with lever control and ball valve act within the individual users. Then, in this context, it could be Proposition 5 For every k, m ∈ N+ we have of great help for the description and interpretation of data and ! dynamics of a complex network, a model that uses blinking m/d νk(m) = μ(d) · k , (28) fractals of order 24 or more, on a daily basis, or of order a d|m multiple of 7 (e.g., 168) on a weekly basis, etc. Similarly, one could think that most of the human activities, within where the sum runs over the positive divisors d of m. the alternation between rest and work, could be studied or Proof approximated by models that use blinking fractals with order The proof is a standard application of Möbius inver- some multiples of 24 and 7 as before, on dependence from sion (cf., for example, Lothaire 1983, Section 1.3) and, in the observations. short, it boils down to recognizing that, fixed k belonging to N+, the sequence km is the Möbius transform of Finally, some conclusions must be also drawn on the rela- m≥1 ν ( ) 6 tionships between our Definition 1 and the paper (Sergeyev k m m≥1 . To give some details recall that, in virtue of 2007) where the idea of cyclicity and periodicity associated Definition 1 and Proposition 1,anm-tuple built with elements with fractals is originally proposed. First of all, in Sergeyev belonging to a set of k distinct simple fractals uniquely iden- (2007) the notion of order of a blinking fractal is not defined tifies a blinking fractal of order a divisor d of m, hence we and, indeed, our Definition 1 is completely inapplicable in have that context. This because all this paper and our discussion ! m = ν ( ). is based on classical mathematics which uses heavily the k k d concept of limit in fact, Sects. 3, 4 and Definition 1 con- d|m sider and investigate the limit behavior, in the usual sense of Using Möbius inversion formula (see, for instance, Apostol fractal geometry, of a sequence of objects in the hyperspace H RN . Instead, on the contrary, in paper (Sergeyev 2007) 1976, Theorem 2.9 or Schroeder 2009, Chap. 21), we then the notion of limit does not exist at all because a different get (28). numerical system is used, and the idea of blinking fractal 5 focuses there on the process that generates a fractal and not The discussion of Sergeyev’s methodology for numerical computations with infinities and infinitesimals goes beyond the aims of this on the limit object such as S, C or Y . In particular, our result article, which instead wants to investigate the possibility of using blinking fractals in classical mathematics and the usefulness of applying on the convergence of Sergeyev’s sequence Sn n∈N does not contradict the discussion in Sergeyev (2007) where the them in the scientific and engineering fields. For more information on sequence maintains its duplicity at infinity in a similar way Sergeyev’s method and its applications in various areas of mathematics, computer science and experimental sciences, we hence refer the to what happens for the sequence sn n∈N defined in (26), reader to Amodio et al. (2017); Antoniotti et al. (2020b,a); Caldarola which, if evaluated at infinity with the methodology used in (2018b); Caldarola et al. (2020a); Cococcioni et al. (2020); Caldarola Sergeyev (2007) gives rise to two opposite infinitesimals (one et al. (2020b); Falcone et al. (2020); Iavernaro et al. (2020); Sergeyev (2013, 2007, 2008, 2009, 2010, 2011, 2016, 2017); Sergeyev and Garro (2010). 6 The Möbius transform is also called, by some authors, the sum-of- divisors transform to not confuse it with the Möbius transformation used in geometry (see, e.g., Weisstein 2002). 123 F. Caldarola, M. Maiolo

Example 3 Using (28) the reader can trivially recover the val- Barnsley MF (2006) Superfractals. Cambridge University Press, Cam- ues in Example 2 or can compute, for instance, the number bridge, UK of blinking fractals of order 12 over the set {F , F , F }, Bernardara P, Lang M, Sauquet E, Schertzer D, Tchiriguyskaia I (2007) 1 2 3 Multifractal analysis in hydrology: application to time series. Édi- obtaining tions Quae, Versailles Cedex Bertacchini F, Bilotta E, Caldarola F, Pantano P, Renteria Bustamante L ! (2016) Emergence of linguistic-like structures in one-dimensional 12/d ν3(12) = μ(d) · 3 cellular automata. In: Sergeyev YD, Kvasov DE, Dell’Accio F, d|12 Mukhametzhanov MS (eds) 2nd Intern. Conf. “NUMTA 2016 - = μ(1) · 312 + μ(2) · 36 + μ(3) · 34 numerical computations: theory and algorithms”, AIP Conf Proc, +μ( ) · 3 + μ( ) · 2 + μ( ) · 1 vol 1776. AIP Publishing, New York, p 090044. https://doi.org/ 4 3 6 3 12 3 10.1063/1.4965408 12 6 4 2 = 3 − 3 − 3 + 3 = 530 640. Bertacchini F, Bilotta E, Caldarola F, Pantano P (2018) Complex interactions in one-dimensional cellular automata and linguis- Acknowledgements The authors wish to thank the anonymous review- tic constructions. Appl Math Sci 12:691–721. https://doi.org/10. ers for their careful reading of the manuscript and especially for their 12988/ams.2018.8353 questions. Bonora M, Caldarola F, Maiolo M (2020) A new set of local indices applied to a water network through Demand and Pressure Driven Funding Open access Funding provided by Università della Calabria Analysis (DDA and PDA). Water (MDPI) 12:2210. https://doi.org/ within the CRUI-CARE Agreement. 10.3390/w12082210 Bonora M, Caldarola F, Maiolo M, Muranho J, Sousa J (2020) The new set up of local performance indices into WaterNet- Compliance with ethical standards Gen and applications to Santarèm’s network. In: Proc. of the 4th international conference on efficient water systems EWaS4, environmental science proceedings, vol 2, p 18. https://doi.org/10. 3390/environsciproc2020002018 Conflict of interest The authors declare that they have no conflict of Bonora M, Caldarola F, Muranho J, Sousa J, Maiolo M (2020) Numer- interest. ical experimentations for a new set of local indices of a water network. In: Sergeyev YD, Kvasov DE (eds) Proceedings of Open Access This article is licensed under a Creative Commons the 3rd international conference numerical computations: theory Attribution 4.0 International License, which permits use, sharing, adap- and algorithms, Lecture Notes in Computer Science, vol 11973, tation, distribution and reproduction in any medium or format, as pp 495–505. 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