December, 2013                                                                                          Microwave Review 
           
                  Fractal Geometry in Electromagnetics Applications  
                                        -  from Antenna to Metamaterials 
                                                                            
                                                          Wojciech J. Krzysztofik 
                                                                            
             Abstract – The effectiveness of antenna and other EM devices        1.   Self-similarity is useful in designing multi-frequency 
          geometry in terms of lowering or establishing a specific resonant           antennas, as, for instance, in the examples based on  
          frequency for different structures of fractal geometry is                   the Sierpinski gasket, and has been applied in 
          considered. We provide a comprehensive overview of recent                   designing of multi-band arrays. 
          developments in the rapidly growing field of modern                    2.   Fractal dimension is useful to design electrically 
          communication, especially mobile systems. This research                     small antennas, such as the Hilbert, Minkowski and 
          revealed unexpected results, which provided additional insight              Koch monopoles or loops, and microstrip patch 
          into unique fractal structures. Results of MoM & FDTD -                     antennas. 
          simulation methods of the circuit- and field antenna and 
          metamaterials parameters in comparison with measurements are           3.   Mass fractals and boundary fractals are useful in 
          presented and discussed.                                                    obtaining high-directivity elements, under-sampled 
             Keywords – Fractal geometry, Multiband antenna, Small                    arrays, and low-sidelobes arrays.  
          printed antenna, Metamaterials, Modern communications.                 4.   Recently, the space-filling Hilbert (Peano) fractal 
                                                                                      curves were used to realize the high-impedance 
                                I.  INTRODUCTION                                      ground planes EBG, so called metamaterials, of the  
                                                                                      high-performance, low-profile, conformal and  flush-
             Antenna design is a mature field of research; it is therefore            mounted antennas with improved radiation 
          rare that a new approach arises in view of the traditional                  characteristics for various communications and radar 
          methods for use into a modern communication systems. In the                 applications. 
          past, antennas had simple form based on Euclidean geometry.          In many EM devices, the self-similarity and plane-filling 
          Recent efforts by several researchers around the world to          nature of fractal geometries are often qualitatively linked to its 
          combine fractal geometry with electromagnetic theory have          frequency characteristics, i.e. multi-frequency operation, or 
          led to a plethora of new and innovative antenna designs.  small size in low frequency bands. 
          Fractal antennas do not follow the Euclidean geometry design.        II.  BRIEF BACKGROUND ON FRACTAL GEOMETRY 
          Their complex structure is built up through replication of a 
          base shape. It has been an intriguing question among 
          electromagnetic community as to what property of fractals, if      A. Fractals in Nature 
          any, is really useful, especially when it comes to designing 
          fractal shaped antenna elements.                                     The original inspiration for the development of fractal 
          Fractals are abstract objects that cannot be physically  geometry came largely from an in-depth study of the patterns 
          implemented. Nevertheless, some related geometries can be          of nature.  
          used to approach an ideal fractal that are useful in 
          constructing antennas. Usually, these geometries are called 
          pre-fractals or truncated fractals. In other cases, other 
          geometries such as multi-triangular or multilevel 
          configurations can be used to build antennas that might 
          approach fractal shapes and extract some of the advantages            a.                                    b.                     
          that can theoretically be obtained from the mathematical 
          abstractions. In general, the term fractal antenna technology is 
          used to describe those antenna engineering techniques that are 
          based on such mathematical concepts that enable one to obtain 
          a new generation of antennas with some features that were 
          often thought impossible in the mid-1980s. 
             After all the work carried out thus far, one can summarize          c.                         d.                           
          the benefits of fractal technology in the following way: 
                                                                             Fig. 1. Fractal objects in nature and technique: fractal cells printed as 
                                                                               metamaterial on the septum of GTEM chamber (a), crystal, snow 
                                                                                        flake (b), the human lungs (c),  fractal art (d) 
               Wojciech  J. Krzysztofik is with the Faculty of Electronic                                    
          Engineering, Wroclaw University of Technology, Wybrzeze              For instance, fractals have been successfully used to model 
                                                                             such complex natural objects as galaxies, cloud boundaries, 
          Wyspianskiego 27, 50-334 Wroclaw, Poland, 
          E-mail: wojciech.krzysztofik@pwr.wroc.pl                           mountain ranges, coastlines, snowflakes, trees, leaves, ferns, 
                                                                                                                                          3 
           Mikrotalasna revija                                                                                               Decembar 2013. 
            
           and much more (Fig. 1). For millions of years of the                      •   have fractal dimension. 
           evolution, nature has been optimizing the architecture of   
           biological structures to effectively distribute and use energy,      B. Why Fractal-Shape Antennas ? 
           and basically a fractal form can be found in every critical 
           structure.                                                             Antennas are essentially narrowband devices. Their 
             Mandelbrot realized [13] that it is very often impossible to       behaviour is highly dependent on the antenna size to the  
           describe nature using only Euclidean geometry that is in terms       operating wavelength ratio. This means that for a fixed 
           of straight lines, circles, cubes, and such like. He proposed        antenna size, the main antenna parameters (gain, input 
           that fractals and fractal geometry could be used to describe         impedance, pattern shape, secondary lobe level, and 
           real objects, such as trees, lightning, river meanders and  distribution) will suffer strong variations when changing the 
           coastlines, to name but a few. Fractal dimension can be non-         operating frequency. The frequency dependence also implies 
           integers, therefore intuitively, we can represent it as a measure    that an antenna has to keep a minimum size relative to 
           of how much space the fractal occupies. Fractals may be  wavelength to operate efficiently. That is, given a particular 
           found in nature or generated using a mathematical recipe. The        frequency, the antenna cannot be made arbitrarily small: it 
           word 'fractal' was coined by Benoit Mandelbrot, sometimes            usually has to keep a minimum size, typically on the order of 
           referred to as the father of fractal geometry, who said “I           a quarter wavelength. These well-known results have been 
           coined  fractal from the Latin adjective fractus. The  constraining for decades the antenna performance in 
           corresponding Latin verb frangere means "to break" to create         telecommunication systems, and they have been the object of 
           irregular fragments. It is therefore sensible - and how  an intensive research with some successful results. However, 
           appropriate for our need ! - that, in addition to "fragmented"       the size to wavelength dependence is still a problem in many 
           (as in fraction or refraction),  fractus should also mean  systems where former antenna designs are not particularly 
           "irregular", both meanings being preserved in fragment” [13].        suitable. In that sense, the fractal design of antennas and 
           Moreover he asked: “Why geometry is often described as  arrays can help in dealing with the problem by contributing 
           ‘cold’ or ‘dry’?  One reason lies in its inability to describe the   with a huge, rich variety of geometrical shapes with some 
           shape of a cloud, a mountain, a coastline, or a tree.  Clouds are    astonishing properties. 
           not spheres, mountains are not cones, coastlines are not 
           circles, and bark is not smooth, nor does lightning travel in a 
           straight line.” 
             To date, there exists no watertight definition of a fractal 
           object. Mandelbrot offered the following definition: “A fractal 
           is by definition a set for which the Hausdorff dimension 
           strictly exceeds the topological dimension”, which he later 
           retracted and replaced with: “A fractal is a shape made of                            a.                               
           parts similar to the whole in some way”. 
             So, possibly the simplest way to define a fractal is as an 
           object which appears self-similar under varying degrees of 
           magnification, and in effect, possessing symmetry across 
           scale, with each small part of the object replicating the 
           structure of the whole. Some examples of self-similarity are 
           shown in Fig. 2. The rectangular outlining indicates a few of 
           the self-similarities of the object. 
                                                  
                          a.                         b.                                  c.     
            Fig. 2. The self-similar components of different fractals: Sierpinski 
                          gasket (a), dragon (b), Koch curve (c) 
                                             
           This is perhaps the loosest of definitions, however, it captures 
           the essential, defining characteristic, that of self-similarity. 
            But here are five properties which most fractals: 
                •   have detail on arbitrarily small scales,                              b.                                            
                •   are usually defined by simple recursive processes, 
                •   are too irregular to be described in traditional              Fig. 3. Why fractal antennas (a), and the various fractal geometry 
                    geometric language,                                          they fall into few main categories: loops, dipoles, multiband fractal 
                •   have some sort of self-similarity                                        patches, antenna arrays, metamaterials (b) 
           4 
           December, 2013                                                                                                Microwave Review 
            
              The reason why the fractal design of antennas and  includes fundamentals about the mathematics, as well as 
           metamaterials appear as an attractive way to make it is few-          studies in fractal antennas and reflections from fractal 
           fold (Fig. 3). First because one should expect a self-similar         surfaces.  
           antenna (which contains many copies of itself at several  The space-filling properties of the Hilbert curve and related 
           scales) to operate in a similar way at several wavelengths.           curves (e.g. Peano fractal) make them attractive candidates for 
           That is, the antenna should keep similar radiation parameters         use in the design of fractal antennas. The space-filling 
           through several bands. Second, because the space-filling  properties of the Hilbert curve were investigated in [8] as an 
           properties of some fractal shapes (the fractal dimension)  effective method for designing compact resonant antennas. 
           might allow fractally shaped small devices to better take  The first four steps in the construction of the Hilbert curve are 
           advantage of the small surrounding space.                             shown in Fig. 4. 
              The fractal design of antennas and arrays results from the         The self-affine fractal geometry [11] presented in Fig 4b is 
           blend of two apparently disjoint disciplines, namely  constructed by scaling a square by a factor of 3 in the 
           electromagnetic theory and geometry. From the early spiral            horizontal direction and by a factor of 2 in the vertical 
           and log-periodic antennas developed in the early sixties by           direction, giving six rectangles, out of which the central 
           Carrel Mayes et al, and from the works of Benoit Mandelbrot           rectangle on the upper side is removed. This is the first 
           on fractal geometry, the fractal antennas appears as natural          iteration. The process is now repeated on the remaining 
           way to explore for multi-frequency operation and for an  rectangles in the second iteration and can be continued ad 
           antenna size reduction.                                               infinitum. This procedure is known as the iterated function 
                                                                                 system (IFS). 
           C. How Fractals can be used as Antennas and why Fractals 
           are Space-filling Geometries                                          D. Iterated Function Systems, IFS: The Language of Fractals 
                                                                                  
              While Euclidean geometries are limited to points, lines,              Iterated function systems (IFS) represent an extremely 
           sheets, and volumes, fractals include the geometries that fall        versatile method for conveniently generating a wide variety of 
           between these distinctions. Therefore, a fractal can be a line        useful fractal structures [1-7], [12-13]. These iterated function 
           that approaches a sheet. The line can meander in such a way           systems are based on the application of a series of affine 
           as to effectively almost fill the entire sheet. These space-          transformations, w, defined by  
           filling properties lead to curves that are electrically very long,     
           but fit into a compact physical space. This property can lead                              wxa bxe                             (1) 
           to the miniaturization of antenna elements.                                                                   
                                                                                                         y  c   dy  f 
           a.                                                                                                            
                                                                                  
                                                                                 or, equivalently, by 
                                                                                  
                                                                                                w(x, y)  (ax  by  e, cx  dy  f ),            (2) 
                                                                                                                                                     
                                                                                 where real number coefficients (a, b, c, d, e, f) are responsible 
           b.                                                                    for movement of fractal element in space:  a, d - scaling, b, c – 
                                                                                 rotation by   ,  angles with respect to axis of coordinating 
                                                                                                1   2 
                                                                                 system, and e, f – linear translation by the vector (e, f) , 
                                                                                 respectively, (see Figure 5). They are expressed as: 
                                                                                   a  cos ;    d  cos ;b sin ; c sin   
                                                                                       1      1        2      2      2     2        1     1
              Fig. 4. Generation the four iterations of Hilbert fractal, the space-
             filling curve (a), and the self-affine fractal multiband antenna (b) 
               
              In the previous section, it was mentioned that pre-fractals 
           drop the complexity in the geometry of a fractal that is not 
           distinguishable for a particular application. For antennas, this 
           can mean that the intricacies that are much, much smaller than 
           a wavelength in the band of useable frequencies can be                                                                             
           dropped out [8]. This now makes infinitely complex structure,                             Fig. 5. The affine transforms 
           which could only be analysed mathematically, but may not be                                              
           possible to be manufactured. It will be shown that the band of        Now suppose we consider w , w , ..., w  as a set of affine 
           generating iterations required to reap the benefits of                                                 1   2       N
           miniaturization is only a few before the additional  linear transformations, and let A be the initial geometry. Then 
           complexities become indistinguishable. There have been  a new geometry, produced by applying the set of 
           many interesting works that have looked at this emerging field        transformations to the original geometry, A, and collecting the 
                                                                                 results from w  (A), w  (A) , …, w (A), can be represented by 
           of fractal electrodynamics. Much of the pioneering work in                            1       2           N
           this area has been documented in [9] and [10]. These works             
                                                                                                                                                  5 
                             Mikrotalasna rrevija                                                                                                                                                                                                                                                                                                  DDecembar 20013. 
                              
                                                                                                                          N                                                                           (3)                                                                                            TABBLE 1 
                                                                                                 W(A)wn(A)                                                                                                               COLLLECTION OF SOOME FRACTAL STTRUCTURES USEEFUL FOR ANTENNNA 
                                                                                                                        n1                                                                                                                                        ANDD METAMATERIAALS APPLICATIOONS 
                             wheere W is knowwn as the Hutchinson operattor [1], [12].                                                                                                                                    
                             A frfractal geomettry can be obttained by repeeatedly applyiing W                                                                                                                           Fractal TType Similarityy                                         Iterated Function System                                     Sketch of Iterated Structurre 
                                                                                                                                                                                                                                                             Dimensionn                               1     1
                             to thhe previous geometry. For example, if thhe set A  repreesents                                                                                                                                                                                         ሺ      ሻ                                                                             
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔ; ݕ൨൨ 
                                                                                                                                                                              0                                                                                                       ଵሺ       ሻ      3     3                                  i = 0      
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ
                             the initial geomettry, then we wwill have                                                                                                                                                                                                                 ଶ 1        √3         1 √√3          1                   
                                                                                                                                                                                                                                                                                    ൌቈ ݔെ ݕ൅ ; ݔ൅ ݕ቉                                            
                                                                                                                                                                                                                                                                                         6         6         3 6            6                  i = 1 
                                                                                                                                                                                                                                                                                        ሺ      ሻ
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ
                                               A W(A ));                             A W(A); ... ; A                                                        W(A )                                  (4)                                                                              ଷ 1        √3         1      √3         1                
                                                  1                      0                2                      1                                   k1                       k                                        Koch Cuurve                         1.2618                 ൌቈ ݔ൅ ݕ൅ ;െെ                         ݔ൅ ݕ                   
                                                                                                                                                                                                                                                                                         6         6         2        6        6               i = 2 
                             An  iterated funnction systemm generates a sequence that                                                                                                                                                                                                   √3൨
                             connverges to a finnal image, A , in such a waay that                                                                                                                                                                                                  ൅ 6               1        2 1
                                                                                                                                                                                                                                                                                      ሺ      ሻ
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔ൅ ; ݕ൨ 
                                                                                                 W(A ))  A                                                                                           (5)                                                                             ସ               3        3 3
                                                                                                                                                                                                                                                                                   
                                                                                                                                                                                                                                                                                        ሺ      ሻ
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ
                             This image is ccalled the atttractor of thee iterated funnction                                                                                                                                                                                          ଵ 1         √3 √3              1
                                                                                                                                                                                                                                                                                    ൌቈ ݔെ ݕ;                   ݔ൅ ݕ቉                            i = 0      
                             systtem, and repreesents a "fixedd point" of W.                                                                                                                                                                                                             2         6       6         2                           
                                                                                                                                                                                                                                                                                        ሺ      ሻ      1         11    1       1
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔ൅ ; ݕ൅ ൨ 
                             Forr  the Koch frfractal curve (Fig. 2c) thee matrix of aaffine                                                                                                                                                                                           ଶ              3        √√3 3          3                  
                                                                                                                                                                                                                                                                                        ሺ      ሻ      1      1       2                           
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔ; ݕ൅ ൨ 
                             trannsformation haas following fform                                                                                                                                                                                                                      ଷ              3      3       3                          i = 1
                                                                                                                                                                                                                         Koch Snoowflake                     2                          ሺ      ሻ      1         11    1       1                  
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔെ ; ݕ൅ ൨ 
                                                                                                                                                                                                                                                                                      ସ               3        √√3 3          3                  
                                                                                                                                                                                                                                                                                        ሺ      ሻ      1         11    1       1                  
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔെ ; ݕെ ൨ 
                                                          x                     cos                                sin               x                t                                                                                                                     ହ
                                                                            q1                 q1                 q2                q2                           q1                               (6)                                                                                             3        √√3 3          3                 i = 2
                                              w                                                                                                                                                                                                                                                 1      1       2
                                                   q                                                                                                                                                                                                                                ሺ      ሻ
                                                                                                                                                                                                                                                                                ݓ ݔ,ݕ ൌ൤ ݔ; ݕെ ൨ 
                                                          y                      sin                      cos                          y                t                                                                                                                     ଺              3      3       3
                                                         q1                                  q1               qq2                q2           q2
                                                                                                                                                                                                                                                                                       ሺ      ሻ      1         11    1       1
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔ൅ ; ݕെ ൨                                         i = 3       
                             andd scaling factor is expressedd as                                                                                                                                                                                                                      ଻              3        √√3 3          3
                                                                                                                                                                                                                                                                                                      1     1                                              
                                                                                                                                                                                                                                                                                        ሺ      ሻ                                                                                   
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔ; ݕ൨൨ 
                                                                                                                                                                                                                                                                                      ଵ               4 14         1 1
                                                                                                                             1                                                                                                                                                          ሺ      ሻ                                                                                                
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤െ ݕ൅ ; ݔ൨ 
                                                                                                 q                                                                                                  (7)                                                                              ଶ                  4        4 4                                           i = 0              i = 1
                                                                                                                                                                                                                                                                                        ሺ      ሻ      1        1 1          1                              
                                                                                                               22cos                                                                                                                                                              ݓ ݔ,ݕ ൌ൤ ݔ൅ ; ݕ൅ ൨ 
                                                                                                                                           q                                                                                                                                           ଷ              4        4 4          4
                                                                                                                                                                                                                                                                                        ሺ      ሻ          1       1 1          1
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤െ ݕ൅ ; ݔ൅ ൨ 
                             wheere     is the  inclination aangle of the second subseection                                                                                                                                                                                         ସ                   4       2 4          4
                                                     qi                                                                                                                                                                  Minkowsski Curve                    1.5                        ሺ      ሻ          1        1 1
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤െ ݕ൅ ; ݔ൨ 
                             withh respect to thhe first, and t  is an elemennt displacemeent on                                                                                                                                                                                       ହ                  4        2 4
                                                                                                                  qqi                                                                                                                                                                   ሺ      ሻ      1        1 1          1
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔ൅ ; ݕെ ൨ 
                             the  plane. Figurre 6  illustratees the iteratedd function syystem                                                                                                                                                                                        ଺              4 1      2 4          4                                                                           
                                                                                                                                                                                                                                                                                        ሺ      ሻ                   3 1         1
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤െ ݕ൅ ; ݔെ ൨ 
                             proccedure for geenerating the wwell-known KKoch fractal ccurve.                                                                                                                                                                                          ଻                  4        4 4         4                                i = 2                i = 3
                                                                                                                                                                                                                                                                                        ሺ      ሻ      1        3 1
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔ൅ ; ݕ൨ 
                             In  tthis case, thee initial set, AA ,  is the linne interval off unit                                                                                                                                                                                    ଼              4        4 4
                                                                                                                       0                                                                                                                                                                              1        1       1       1
                                                                                                                                                                                                                                                                                       ሺ      ሻ                                                           
                                                                                                                                                                             1                                                                                                      ݓ ݔ,ݕ ൌ൤ ݕെ ;െ ݔെ ൨ 
                             lenggth, i.e.,                                                                            ,  =60 , andd                                             .                                                                                                   ଵ               2        2       2       2
                                                                                                                   
                                                                A  x:                         x[0,1]]                       q                                                                                                                                                                     1        1 1          1
                                                                   0                                                                                               q                                                                                                                    ሺ      ሻ
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔെ ; ݕ൅ ൨ 
                                                                                                                                                                             3                                                                                                         ଶ              2        2 2          2
                                                                                                                                                                                                                         Hilbert CCurve                      1.2619                     ሺ      ሻ      1        1 1          1
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔ൅ ; ݕ൅ ൨ 
                                                                                                                                                                                                                                                                                       ଷ              2        2 2          2                                                                               
                                                                                                                                                                                                                                                                                                          1       1 1          1                           
                                                                                                                                                                                                                                                                                        ሺ      ሻ                                                   i = 0              i = 1          i = 2 
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤െ ݕെ ; ݔെ ൨ 
                                                                                                                                                                                                                                                                                      ସሺ       ሻ      1 2 1       2 2          2
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔ; ݕ൨൨ 
                                                                                                                                                                                                                                                                                      ଵ               2     2
                                                                                                                                                                                                                                                                                        ሺ      ሻ      1        1 1
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔ൅ ; ݕ൨ 
                                                                                                                                                                                                                         Sierpinskki Gasket                  1.5849                    ଶ              2        2 2                                                                                        
                                                                                                                                                                                                                                                                                        ሺ      ሻ      1        1 1          √3                             
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌቈ ݔ൅ ; ݕ൅ ቉ 
                                                                                                                                                                                                                                                                                      ଷ               2        4 2           4                    i = 0              i = 1          i = 2 
                                                                                                                                                                                                                                                                                        ሺ      ሻ      1     1                                                                      
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔ; ݕ൨൨ 
                                                                                                                                                                                                                                                                                      ଵ               3     3                                                                      
                                                                                                                                                                                                                                                                                        ሺ      ሻ      1      1       1                                                             
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔ; ݕ൅ ൨ 
                                                                                                                                                                                                                                                                                       ଶ              3      3       3                                                             
                                                                                                                                                                                                                                                                                        ሺ      ሻ      1      1       2
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔ; ݕ൅ ൨ 
                                                                                                                                                                                                                                                                                       ଷ              3      3       3                                                             
                                                                                                                                                                                                                                                                                         ሺ      ሻ      1        11  1
                                                                                                                                                                                                                                                                                     ݓ ݔ,ݕ ൌ൤ ݔ൅ ; ݕ൨ 
                                                                                                                                                                                                                                                                                       ସ               3        33  3
                                                                                                                                                                                                                         Sierpinskki Carpet                  1.8927                     ሺ      ሻ      1        1 1          2
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔ൅ ; ݕ൅ ൨ 
                             a.                                                                                                                                                                                                                                                        ହ              3        3 3          3                                                                              
                                                                                                                                                                                                                                                                                        ሺ      ሻ      1        2 1                                                                 
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔ൅ ; ݕ൨ 
                                                                                                                                                                                                                                                                                       ଺              3        3 3                                 i == 0              i = 1          i = 2 
                                                                                                                                                                                                                                                                                        ሺ      ሻ      1        2 1          1
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔ൅ ; ݕ൅ ൨ 
                                                                                                                                                                                                                                                                                       ଻              3        3 3          3
                                                                                                                                                                                                                                                                                        ሺ      ሻ      1        2 1          2
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔ൅ ; ݕ൅ ൨ 
                                                                                                                                                                                                                                                                                      ଼               3        3 3          3
                                                                                                                                                                                                                                                                                                     ሾ                       ሿ                 
                                                                                                                                                                                                                                                                                        ሺ      ሻ                                               
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ 0.382ݔ;;0.382ݕ  
                                                                                                                                                                                                                                                                                      ଵሺ       ሻ                                               
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ
                                                                                                                                         2 / 3                                                                                                                                         ଶ
                                                                                                                                                                                                                                                                                        ሾ                                    ሿ
                             b.                                                                                                                                                                                                                                                     ൌ 0.382ݔ൅0.618;0.382ݕ  
                                                                                                                                                                                                                                                                                        ሺ      ሻ
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ
                                                                                                                                                                                                                                                                                       ଷ
                                                                                                                                                                                                                                                                                        ሾ
                                                                                                                                                                                                                                                                                    ൌ 0.382ݔ൅0.809;0.382ݕ
                                                                                                                                                                                                                         Sierpinskki Pentagon                1.6722                 ൅0.588ሿ 
                                                                                                                                                                                                                                                                                        ሺ      ሻ
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ                                                                                                                  
                                                                                                                                                                                                                                                                                      ସ                                                        
                                                                                                                                                                                                                                                                                        ሾ
                                                                                                                                                                                                                                                                                    ൌ 0.382ݔ൅0.309;0.382ݕ                                          i = 0              i = 1          i = 2 
                                                                                                                                                                                                                                                                                    ൅0.951ሿ 
                                                                                                                                                                                                                                                                                        ሺ      ሻ
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ
                                                                                                                                                                                                                                                                                       ହ
                                                                                                                                                                                                                                                                                        ሾ                         0.382ݕ
                                                                                                                                                                                                                                                                                    ൌ 0.382ݕെ0.191;
                                                                                                                                                                                                                                                                                    ൅0.588ሿ 
                                                                                                                                                                                                                                                                                      
                                                                                                                                                                                                                                                                                                      ଵ     ଵ
                                                                                                                                                                                                                                                                                        ሺ      ሻ
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌቂݔ; ݕ൅൅2ቃ;  
                                                                                                                                                                                                                                                                                      ଵ               ଷ     ଷ
                                                                                                                                                                                                                         Fractal TTree                       1.5849                     ሺ      ሻ      1        4 1
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔെ ; ݕ൅2൨ 
                                                                                                                                                                                                                                                                                       ଶ              3        3 3
                                                                                                                                                                                                                                                                                        ሺ      ሻ      ଵ        ସ ଵ
                             c.                                                                                                                                                                                                                                                     ݓ ݔ,ݕ ൌቂݔ൅ ; ݕ൅2ቃ 
                                                                                                                                                                                                                                                                                       ଷ              ଷ        ଷ ଷ                         i = 0    i = 1       i = 2          i == 3                  
                                                                                                                                                                                                                                                                                                                                                           
                               Figg. 6. The fractal Koch curve as an iterated funnction system (aa), the                                                                                                                                                                                              ଵ     ଵ                                              
                                                                                                                                                                                                                                                                                        ሺ      ሻ
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌቂݔ; ݕെെܹቃ 
                                                                                                                                                                                                                                                                                      ଵ               ଷ     ଷ                             W 
                                      affine transformmation matricess (b), and the firrst 4-stages in thhe                                                                                                              Cantor Set                          0.6309                     ሺ      ሻ      1        2 1
                                                                                                                                                                                                                                                                                    ݓ ݔ,ݕ ൌ൤ ݔ൅ ; ݕെܹ൨ 
                                                                                            constructtion of it (c)                                                                                                                                                                    ଶ              3        3 3
                                                                                                                                                                                                                                                                                                                                               
                             6