AAbstract Fractals
Marat Akhmet ∗ and Ejaily Milad Alejaily Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey
We develop a new definition of fractals which can be considered as an abstraction of the fractals determinedthrough self-similarity. The definition is formulated through imposing conditions which are governed the relationbetween the subsets of a metric space to build a porous self-similar structure. Examples are provided to confirmthat the definition is satisfied by large class of self-similar fractals. The new concepts create new frontiers for fractalsand chaos investigations.
Fractals are class of complex geometric shapes with certain properties. One of the main features of the objectsis self-similarity which can be defined as the property whereby parts hold similarity to the whole at any level ofmagnification [3]. Fractional dimension is suggested by Mandelbrot to be a property of fractals when he defined afractal as a set whose Hausdorff dimension strictly larger than its topological dimension [2]. Roots of the idea ofself-similarity date back to the 17th century when Leibniz introduced the notions of recursive self-similarity [9]. Thefirst mathematical definition of a self-similar shape was introduced in 1872 by Karl Weierstrass during his studyof functions that were continuous but not differentiable. The most famous examples of fractals that display exactself-similarity are Cantor set, Koch curve and Sierpinski gasket and carpet which where discovered by Georg Cantorin 1883, Helge von Koch in 1904 and Waclaw Sierpinski in 1916 respectively. Julia sets, discovered by Gaston Juliaand Pierre Fatou in 1917-19, gained significance in being generated using the dynamics of iterative function. In1979, Mandelbrot visualized Julia sets including the most popular fractal called Mandelbrot set.In this paper we introduce a new mathematical concept and call it abstract fractal. This concept is an attemptto establish a pure foundation for fractals by abstracting the idea of self-similarity. We define the abstract fractal asa collection of points in a metric space. The points are represented through an iterative construction algorithm withspecific conditions. The conditions are introduced to governor the relationship between the sets at each iteration.Our approach of construction is based on the concept of porosity rather than the roughness notion introduced byMandelbrot. Porosity is an intrinsic property of materials and it is usually defined as the ratio of void volume tototal volume [10]. The concept of porosity plays an important role in several fields of research such as geology,soil mechanics, material science, civil engineering, etc. [10, 11]. Fractal geometry has been widely used to studyproperties of porous materials. However, the concept of porosity was not utilized as as a criterion for fractalstructures, and the relevant researches have investigated the relationship between porosity and fractalness [12–16].For instance several researches such as [17–19] determined fractal dimension of some pore-structures using the poreproperties of them. The simplicity and importance of the porosity concept insistently invite us to develop a newdefinition of fractals through porosity. In other words the property should be involved in fractal theory as a featureto be equivalent to self-similarity and fractional dimension. This needs to specify the concept of porosity to surfacesand lines. In the present paper, we do not pay attention to equivalence between the definition of fractals in termsof porosity and those through self-similarity and dimension rather we introduce an abstract definition which, wehope, to be useful in application domains.
In this paper we shall consider the metric measure space defined by the triple (
X, d, µ ), where (
X, d ) is a compactmetric space, d is a metric on X and µ is a measure on X .To construct abstract fractal, let us consider the initial set F ⊂ X and fix two natural numbers m and M suchthat 1 < m < M . We assume that there exist M nonempty disjoint subsets, F i , i = 1 , , ...M , such that F = ∪ Mi =1 F i . ∗ Corresponding Author Tel.: +90 312 210 5355, Fax: +90 312 210 2972, E-mail: [email protected] a r X i v : . [ m a t h . M G ] A ug or each i = 1 , , ...m , again, there exist M nonempty disjoint subsets F ij , j = 1 , , ...M such that F i = ∪ Mj =1 F ij .Generally, for each i , i , ..., i n , i k = 1 , , ...m , there exist M nonempty disjoint sets F i i ...i n j , j = 1 , , ...M , suchthat F i i ...i n = ∪ Mj =1 F i i ...i n j , for each natural number n . The following conditions are needed:There exist two positive numbers, r and R , such that for each natural number n we have r ≤ (cid:80) mj =1 µ (cid:0) F i i ...i n − j (cid:1)(cid:80) Mj = m +1 µ (cid:0) F i i ...i n − j (cid:1) ≤ R. (1)where i k = 1 , , ..., m, k = 1 , , ...n −
1. We call the relation (1) the ratio condition . The numbers r and R in (1)are characteristics for porosity. Another condition is the adjacent condition and it is formulated as follows:For each i i ...i n , i k = 1 , , ..., m there exists j, j = m + 1 , m + 2 , ..., M , such that d ( F i i ...i n , F i i ...i n − j ) = 0 . (2)We call F i i ...i n − i n a complement set of order n if i k = 1 , , ..., m, k = 1 , , ...n − i n = m + 1 , m + 2 , ..., M .An accumulation point of any couple of complement sets does not belong to any of them. We dub this stipulationthe accumulation condition .Let us define the diameter of a bounded subset A in X by diam( A ) = sup { d ( x , y ) : x , y ∈ A } . Considering theabove construction, we assume that the diameter condition holds for the sets F i i ...i n , i.e.,max i k =1 , ,...M diam( F i i ...i n ) → n → ∞ . (3)Fix an infinite sequence i i ...i n ... . The diameter conditions as well as the compactness of X imply that there existsa sequence ( p n ), such that p ∈ F , p ∈ F i , p ∈ F i i , ... , p n ∈ F i i ...i n , n = 1 , , ... , which converges to a pointin X . The points are denoted by F i i ...i n ... .We define the abstract fractal F as the collection of the points F i i ...i n ... such that i k = 1 , , ...m , that is F = (cid:8) F i i ...i n ... | i k = 1 , , ...m (cid:9) , (4)provided that the above four conditions hold. The subsets of F can be represented by F i i ...i n = (cid:8) F i i ...i n i n +1 i n +2 ... | i k = 1 , , ..., m (cid:9) , (5)where i i ...i n are fixed numbers. We call such subsets subfractals of order n . In this section we find the pattern of abstract fractal in some geometrical well-known fractals, Sierpinski carpet,Pascal triangles and Koch curve.
To construct an abstract fractal corresponding to the Sierpinski carpet, let us consider a square as an initialset F . Firstly, we divide F into nine ( M = 9) equal squares and denote them by F i , i = 1 , , ... F i , i = 1 , , ... F ij , j = 1 , , ...
9. Figure 1 (b) illustrates the sub-squares of F . We continue in this way such that at the n th step, each set F i i ...i n − , i k = 1 , , ...
8, is divided into nine subset F i i ...i n − j , j = 1 , , ...
9. For the Sierpinskicarpet the number m is 8, and the measure ratio (1) can be evaluated as follows. If we consider the first order sets F i , i = 1 , , ...
9, then (cid:80) j =1 µ (cid:0) F j (cid:1) µ (cid:0) F (cid:1) = 8 . Thus, the ratio condition holds. From the construction, we can see that each F i i ...i n , i k = 1 , , ..., F i i ...i n − j , j = 9. Therefore, the adjacent condition holds. Since the construction consistsof division into smaller parts, the diameter condition is also valid. Moreover, It is clear that the accumulationcondition holds as well.As a result, the points of the desired abstract fractal F can be represented as F i i ...i n ... and the abstractSierpinski carpet is defines by F = (cid:8) F i i ...i n ... | i k = 1 , , ... (cid:9) . F and illustrates its 1 st order subfractals. (a) (b) (c) Figure 1: Sierpinski carpet
Pascal triangle is a mathematical structure consists of triangular array of numbers. Triangular fractals can beobtained if these numbers are plotted using specific moduli. The Sierpinski gasket, for instance, is the Pascal’striangle modulo 2. Let us build an abstract fractal on the basis of a fractal associated with Pascal triangle modulo3. Consider an equilateral triangle as an initial set F . In the first step, we divide F into nine smaller equilateraltriangles ad denote them as F i , i = 1 , , ..., F i , i = 1 , , ..., F ij , j = 1 , , ...,
9. Figure 2 (a) illustrates the second step for the set F .Similarly, the subsequent steps are performed such that at the n th step, each set F i i ...i n − , i k = 1 , , ...
6, is dividedinto nine subset F i i ...i n − j , j = 1 , , ...
9. In this case we have m = 6 and M = 9. Therefore, (cid:80) j =1 µ (cid:0) F j (cid:1)(cid:80) j =7 µ (cid:0) F j (cid:1) = 2 , and the ratio condition holds. One can also verify that the adjacent, the accumulation, and the diameter conditionsare also valid. Based on this, the points of the fractal can be defined by F i i ...i n ... , and thus, the abstract Pascaltriangle is defined by F = (cid:8) F i i ...i n ... | i k = 1 , , ... (cid:9) , and the n th order subfractals can be written as F i i ...i n = (cid:8) F i i ...i n i n +1 ... | i k = 1 , , ..., (cid:9) , where i i ...i n are fixed numbers. (a) (b) (c) Figure 2: Pascal triangle modulo 33 .3 The Koch Curve
In this subsection, we shall show how to build an abstract fractal F conformable to the Koch curve. For this purpose,we consider the following construction of the Koch curve. Start with with an isosceles triangle F with base angles of30 ◦ . The first step of the construction consists in dividing F into three equal-area triangles F , F and F (see Fig.3 (b)). The triangles F and F are isosceles with base angles of 30 ◦ , whereas the central triangle F is an equilateralone. In the second step, each F i , i = 1 , F i and F i , and oneequilateral, F i . Figure 3 (c) illustrate the step. In each subsequent step, the same procedure is repeated for eachisosceles triangles resulting from the preceding step. That is, in the n th step, each F i i ...i n − , i k = 1 ,
2, is dividedinto three parts, two isosceles triangles F i i ...i n − j , j = 1 ,
2, with base angles of 30 ◦ , and one equilateral triangle F i i ...i n − . In this construction, we have m = 2 and M = 3, thus, the measure ratio is µ ( F ) + µ ( F ) µ ( F ) = 2 , and the ratio condition holds. From the construction, it is clear that the adjacent, the accumulation, and thediameter conditions are also valid. Based on this, the points in F can be represented F i i ...i n ... , and thus, theabstract Koch curve is defined by F = (cid:8) F i i ...i n ... | i k = 1 , (cid:9) . (a) (b)(c) Figure 3: Abstract Koch curve constructionThe n th order subfractals of F are represented by F i i ...i n = (cid:8) F i i ...i n i n +1 i n +2 ... | i k = 1 , (cid:9) , (6)where i i ...i n are fixed numbers. Figure illustrats examples of 1 st , nd , rd and 4 th subfractals of the abstract Kochcurve. Figure 4: Subfractals of the abstract Koch curve4 Abstract self-similarity and Chaos
In paper [1], we have introduced the notion of the abstract self-similarity and defined a self-similar set by F = (cid:8) F i i ...i n ... : i k = 1 , , ..., m, k = 1 , , ... (cid:9) , (7)where F i i ...i n ... , i k = 1 , , ..., m represent the points of the set. For fixed indexes i , i , ..., i n , the subsets areexpressed as F i i ...i n = (cid:91) j k =1 , ,...,m F i i ...i n j j ... , (8)such that F i i ...i n = ∪ mj =1 F i i ...i n j , for each natural number n , where all sets F i i ...i n j , j = 1 , , ..., m , arenonempty, disjoint and satisfy the diameter condition.Based on the definition of the abstract self-similar set, we see that every abstract fractal is an abstract self-similarset, but the reverse is not necessarily valid.A similarity map ϕ for the abstract fractal F can be defined by ϕ ( F i i i ... ) = F i i i ... . Let us assume that the separation condition holds, that is, there exist a positive number ε and a natural number n such that for arbitrary i i ...i n one can find j j ...j n so that d (cid:0) F i i ...i n , F j j ...j n (cid:1) ≥ ε , where ε is the separation constant. Considering the results on chaos for self-similar set provided in [1], it canbe proven that the similarity map ϕ possesses the three ingredients of Devaney chaos, namely density of periodicpoints, transitivity and sensitivity. Moreover, ϕ possesses Poincar`e chaos, which characterized by unpredictablepoint and unpredictable function [7, 8]. In addition to the Devaney and Poincar`e chaos, it can be shown that theLi-Yorke chaos also takes place in the dynamics of the map. These results are summarized in the next theoremwhich can be proven in the similar way that explained in [1]. Theorem 1.
If the separation condition holds, then the similarity map possesses chaos in the sense of Poincar´e,Li-Yorke and Devaney..
That is the triple ( F , d, ϕ ) is a self-similar space and ϕ is chaotic in the sense of Poincar´e, Li-Yorke and Devaney. Iterated function system (IFS) is a powerful tool for the construction of fractal sets. It is defined by a family ofcontraction mappings w n , n = 1 , , ... N on a complete metric space ( X, d ) [4,5]. The procedure starts with choosingan initial set A ∈ B ( X ), where B ( X ) is the space of the non-empty compact subsets of X , then iteratively applyingthe map W = { w n , n = 1 , , ... N } such that A k +1 = W ( A k ) = (cid:83) Nn =1 A nk , where A nk = w n ( A k ). The fixed point ofthis map, A = W ( A ) = lim k →∞ W k ( A ) ∈ B ( X ), is called the attractor of the IFS which represents the intendedfractal.The idea of the structure of the abstract fractal can be realized using IFS. The fractal constructed by anIFS is an invariant set. Therefore, the subsets at each step of constructions can be determined using the maps w n , n = 1 , , ... m as illustrated if Fig. 5 (a). Similarly, the maps transform each subfractal into subfractals of thesubsequent order. Figure 5 (b) demonstrates the action of w n ’s on the abstract fractal. The difference between thiscase and the above IFS fractal construction is that the sets F i i ...i n are fractals in themselves, whereas the sets A nk are not.Utilizing the idea, moreover, each subfractal can be expressed in terms of the iterated images of whole fractal F , that is F i = w i ( F ) , F ij = w j ( w i ( F )) , F ijk = w k ( w j ( w i ( F ))) , ... , thus, in general we have F i i ...i n = w i n ( w i n − ( ... ( w i ( F ))) ... ) , from which we can define a point belong to the fractal as the limit of the iterated images of F , F i i ... = lim n →∞ w i n ( w i n − ( ... ( w i ( F ))) ... ) . ε can be expressed in terms of w n such that the condition is satisfied ifmin n inf i n ,j n =1 , ,... m d (cid:0) w i n ( ... ( w i ( F )) , w j n ( ... ( w j ( F )) (cid:1) ≥ ε . (a) (b) Figure 5: IFSIn addition to the construction of fractals, the IFS is used to prove chaos for the so-called totally disconnectedIFS corresponding to certain classes of self-similar fractals like the Cantor set [6]. The proof consists of constructionof a dynamical system { A ; S } , where S : A → A is the shift transformation defined by S ( a ) = w − n ( a ) for a ∈ W n ( A ).The system is called the shift dynamical system associated with the IFS, and then it showed to be topologicallyconjugate to the shift map on the N -symbols code space. We see that this approach follows the usual constructionof chaos which begins with defining a map with certain properties where the conjugacy to a well known chaoticmap is the major key in discovering the chaotic nature of fractals. Again we emphasize that this approach is onlyapplicable for the totally disconnected fractals namely the well-known Cantor sets. Differently, our approach ischaracterized by the similarity map ϕ : F → F which, with regard to IFS approach, can be seen as an abstractionof the geometric essence of the transformation S . Using the idea of indexing the domain elements allows to definethe abstract map ϕ . This has shortened the way of chaos proving by eliminating the need for topological conjugacy.Moreover, it becomes possible to investigate the chaotic nature in several classes of fractals such as the Sierpinskifractals and the Koch curve. The fractal concept is axiomatically linked with the notion self-similarity. This is why it is considered to be oneof the two acceptable definitions of fractals. That is, a fractal can be defined as a set that display self-similarityat all scales. Mandelbrot define a fractal as a set whose Hausdorff dimension strictly larger than its topologicaldimension. In the present research, we introduce a conception of abstract fractal which can be considered as anothercriterion of fractalness. Indeed, the idea of the abstract fractal centers around the self-similarity property and manyself-similar fractals like the Cantor sets and the Sierpinski fractals are shown to be fractals in the sense of theabstract fractal. These fractals are also satisfied the Mandelbrot definition. Moreover, in our previous paper [1],we have also shown that the set of symbolic strings satisfies the definition of abstract self-similarity. Because ofthese facts, we believe that the notion of abstract fractal deserves to be the third definition of fractals and we hopeit will be accepted by the mathematical community. Considering the abstract fractal as a new definition of fractalmay open new opportunities for more theoretical investigations in this field as well as new possible applicationsin science and engineering. For example, we may start with the equivalency between these definitions. It isknown that the fractals that display exact self-similarity at all scales satisfy Mandelbrot definition of fractal. Theproposed definition satisfies the self-similarity since it is the main pivot of the concept of the abstract fractal. Butthe interesting question is: Does the abstract fractal agree with Mandelbrot definition? The notion of Hausdorffdimension for the abstract fractal is not yet developed enough to provide an answer to the question. However, thedetermination of the fractal dimension can possibly be performed based on two important properties. The firstone is the self-similarity of the abstract fractal which may provide a self-similar dimension that can be assumed to6e equivalent to the Hausdorff dimension. The second one is the accumulation condition combining perhaps withthe diameter condition. These properties are essential for describing the geometry of fractals, therefore, the fractaldimension can be characterized in terms of them. This is why the definition in our paper can give opportunitiesto compare abstract fractals with fractals defined through dimension. Furthermore, the suggested fractal definitioncan be elaborated through chaotic dynamics development, topological spaces, physics, chemistry, neural networktheories development.
References [1] Akhmet, M. and Alejaily E. M. 2018, Abstract Similarity, Chaos and Fractals. (submitted)[2] Mandelbrot, B. B. 1983, The Fractal Geometry of Nature, Freeman, New York.[3] Addison, P. S. 1997, Fractals and Chaos: An Illustrated Course, Institute of Physics Publishing, Bristol, UK.[4] Hutchinson J. 1981, Fractals and self-similarity. Indiana Univ. J. Math.399