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Fractal Geometry and Its Application to Antenna Designs

This paper primarily focuses on various fractal geometries and their applications to antenna designs. Several natures inspired and human inspired fractal geometries are presented one by one. Their importance and design procedure are also briefly discussed. The dimensions of such fractal geometries are found using their mathematical modeling. Considering modeling and their corresponding shapes various low profiles, low cost, small size and, light weight antenna designs for various wireless applications are described. The broadband, wideband, and multiband nature of the design due to fractal application are discussed. Finally advantages, disadvantages, major applications, and future scope of such fractal geometries are mentioned.


INTRODUCTION
Now a day, the wireless communication devices become portable due to implementation of the recent technologies. Antenna is a major element used in communication devices. Hence, antenna minimization is the latest research topic for many researchers. Antenna minimization depends on the relationship between the physical sizes of the antenna with its operational wavelength. This relationship is a major and most essential parameter in the area of antenna design. The physical size of antenna element is inversely proportional to its operational frequency. But when the physical size of antenna is reduced then its electrical size also reduces as there is no change in the operational frequency. Again the electrical size of antenna is expressed in terms of the operational wavelength (λ). The physical size of an antenna is normally considered as the half or quarter of its operational wavelength. The antenna operates satisfactorily over the range of frequencies called as the bandwidth which is generally 10-40% of the center wavelength. But when the dimensions of the antenna become much smaller than its operating wavelength then it reduces the radiation resistance, S11 parameters, bandwidth radiation performance and efficiency of the antenna. Some common examples of antennas with the quarter-wavelength of the electrical size are monopole antennas, helical antennas and planar inverted-F antennas (PIFAs) [1].
The fractal geometries have wide applications in the field of biology, geography and engineering. In the field of engineering, fractal geometries have been used in the process of antenna designs, frequency selective surface designs, image processing and bio medical signal processing. The concept of fractal antenna theory is a relatively new research area in the field of antenna design. But due to various attracting features the Fractal antennas and the corresponding superset fractal electrodynamics is a major attraction of current research activity [2]. Fractal geometries are considered as the complex geometric shapes with self-similarity, self-scaling and space-filling properties. These properties make them a suitable candidate in miniaturized antenna designs. The Space-filling property results in electrically large size features. Self-similar property enables use of iteration function system with similar shapes. Self-scaling property allows iteration function system to use similar shapes of multiple scales. These features enable them to be efficiently packed, thus easily represented into small areas. The antenna miniaturization process can be achieved through the implementation of self scaling, space filling and self similarity properties of fractals that produces the curves which are electrically very long with a compact structured physical space [3]. Due to self-similarity, self scaling and space filling properties, fractal geometries are widely used in Fractal antenna designs. When Fractal antennas are compared with the conventional antenna, then it is found that the fractal antennas have much greater bandwidth with very compact antenna size. By using the fractal antennas multiple resonant frequencies can be achieved which are multiband but are not harmonics in nature [4]. Hence, antenna designs based on fractal geometries are suitable for various wireless applications. The theoretical and conceptual foundations of antennas were laid on famous Maxwell's equations. The Scottish scientist James Clark Maxwell observed the theories of electricity and magnetism in 1873 and eventually represented their relationship through a set of mathematical equations called as Maxwell's Equations. And in 1886 German scientist Heinrich Rudolph Hertz verified the Maxwell's Equations and invented that the electrical disturbances could be detected with a secondary circuit of particular dimensions for resonance and contains an air gap for occurrence of sparks [5]. The Italian scientist Guglielmo Marconi designed a microwave device of parabolic cylindrical shape at a particular wavelength of 25 cm for his original code transmission and further worked at larger wavelengths for improvement in the communication range. Hence the Marconi is regarded as the "father of amateur radio". In the early years the antenna developments were limited by the availability of signal generators. But in 1920, the resonant length antennas were invented when the De Forest triode tube was introduced to produce the continuous wave signals ranging up to 1MHz [6]. The term 'fractal' means wrecked or broken or irregular segments. This term 'fractal' was invented by the French scientist Dr. B Mandelbrot in 1975. The term 'fractal' was first derived from the word frangee (i.e. a Greek word) and the word fractus (i.e. a Latin word) which means wrecked or broken or irregular segments. Dr. B Mandelbrot investigated that there exists a fundamental relationship between the fractal dimensions and the nature patterns that exist in nature [7]. In 1988, the first type of the fractal antenna was designed by Canadian scientist Dr. Nathan Cohen. And he has suggested that the Fractal antenna is a new type of antenna which is simple and easy to manufacture, follows self-similar and self-repetitive characteristic, thus could be suitable in military as well as commercial applications. Dr. Nathan Cohen has introduced the new concept on fractalization of various geometries of a dipole or loop antenna initially. This concept suggests in bending of a wire in such a fractal way that the entire length of the particular antenna remains the same but the antenna size is reduced with the addition process of continuous iterations. When this fractal concept is properly implemented then an efficient technique of miniaturized antenna design is possible. Dr. Nathan Cohen compared the perimeter of a particular Euclidean antenna with a fractal shaped antenna and stated that the fractal antenna follows a perimeter which is not directly proportional to the antenna area. Also he has concluded that in multi-iteration fractal geometry the antenna area is smaller than an Euclidean shaped antenna [8][9]. The concept of fractal dimension is very old. Several fractal geometries are inspired from nature, human. These nature and human inspired fractal geometries are widely used in various science and engineering fields. Fractal geometries are characterized in terms of their corresponding dimensions. The fractal dimensions are determined using their mathematical modeling. The mathematical modeling are specified in the form of mathematical expressions. Various shapes based on these mathematical expressions are used to design small size, low profile, low cost, and light weight antenna designs. Fractal geometries provide multiband, wideband, and broadband nature in the antenna designs. Fractal geometries are currently major cause of interest for various researchers in the field of science and engineering due to their key advantages. Hence, these make fractal geometries a suitable candidate in wireless communication for various major applications.
Our research goal is to discuss the concept and various applications of fractal geometry to antenna structures. The revolution of fractal geometry is shown in section 2. Section 3 represents the mathematical modeling of various fractal geometries. The detailed literature review is done in section 4. Section 5 contains the overall discussion of major features of fractal geometry. The conclusion of paper is presented in the section 6. Eventually, the future work of this work is presented in section 7.

II. REVOLUTION OF FRACTAL GEOMETRY
The term "fractal" may be described as any geometric object i.e. it may be a line or a rectangle or a circle which is irregular or rough in terms of length or size. So, it looks like a 'broken up' shape in a systematic and thorough way. In fractal geometry, the original object is sub-divided into several individual parts where each part is very similar to the original one. This property is called as self-similar property which is occurring at various stages of magnification. In fractal geometry, the original object is scaled with different dimensions which are called as scaling property. So the natural objects are usually self-similar that makes fractal structures suitable in field of antenna design [10].

Fractal Geometry: Inspiration through nature to human
Various fractal structures are inspired from the nature. Following are some typical example of fractal structures that are inspired through nature to human body. We know that the Earth is the only one planet of the solar system where the Life is possible. There are four multiple layers present inside the earth. Here the fractal shown in the following figure represents the super formula which is near c=0 with 215 iterations. Fig.1: The Earth and its corresponding fractal shape. The Egyptian Pyramids were constructed following the images of the star's positions in the sky. Hence there lies a correlation between the earth and the sky. So we may assume that the pyramid shapes were found as the first similarity with the fractal structures. And the Sierpinski gasket fractal antenna structure is very similar to the Egyptian pyramids. There are some vegetables like Cauliflowers and broccoli that possess like a fractal tree-shaped typical structure. Here the fractal geometry is designed by using an if and else equation and the individual branches on each stems are originated by using the power terms of the factors like (z n +1) or (z n -1).

Fig.3: The Fungus and its corresponding fractal shape
We know that a tree structure is the simplest example of fractal dimension in the living world of the nature from the biology. Here a mathematical formula is stated below that generates this fractal form. And the iteration numbers are related to the branch numbers. z n+1 = (z n +1)/c or z n+1 = (z n -1)/c for x>0 and x<0 respectively

Fig.4: The Tree and its corresponding fractal shape
We know that the nerve cells or Neurons are the cells those are electrically excitable belonging to nervous system used for the processing of the transmitting information. And the neurons are mainly combination of a cell body called as soma which is a dendrite tree with an axon. Here the used fractal formula follows the form of if and else formula which is the combination of the terms like sin(n)-1 and sin(n)+1 .

Fig.5: The Neurons and its corresponding fractal shape
Continuous scale-invariance is a property containing un particles but not particles those are used to interpret by fractal dimensions of iteration of various complex functions. Koch Curve is a such type of fractal structure which is a case of discrete scaled invariance property as it remains the same when multiplied with a constant number.

c) The Cantor Set as Fractal geometry
The Cantor Set as fractal geometry is generated by cutting a single line from its centre repeatedly as shown in the following figure.  The Cantor Curtains as fractal geometry is generated by making a gap along the single line from its centre and the process is repeatedly for the multiple iterations as shown in the following figure. The Cantor Square as fractal geometry is generated by taking a Plus shaped structure as the basic shape. Again four numbers of plus shaped structure is created in second iteration and sixteen numbers of squares are created and this procedure continues for further iterations as shown in the following figure. The Cantor Square as fractal geometry is generated by making four triangular slots of a square along midpoint of each side. Then the same process is repeated for multiple iterations as shown in the following figure.

Fig.13: Example of fractal geometry made up by the Cesaro Sweep h) Vicsek snowflake-box fractal
The Vicsek snowflake-box fractal as fractal geometry is generated by taking a square as the basic shape. In the first iteration four numbers of squares are placed at each corners of the square and the process is repeated for multiple iterations as shown in the following figure.

k) The Sierpinski Pyramid as Fractal geometry
The Sierpinski Pyramid as fractal geometry follows a triangular shape and generates a three dimensional structural space of a form of pyramidal shape as shown in the following figure. The star fractal as fractal geometry follows a star shape structure with self similarity property in a particular direction. This repetition process results in generation of similar structures from the previous iteration. Fractal geometries using the Iterated Function System (IFS) method are generated based on the simple analytical methods of plane transformations like up scaling method, down scaling method, Dislocation of objects and the plane of axes rotation methods. The IFS fractal geometry creation consists of the following steps [11] as followes.
Step 1: Defines a set of planes that can be used for transformations.
Step 2: Draws any initial geometry pattern on the plane.
Step 3: The transformation of the initial geometry pattern using the transformation process stated in the first step.
Step 4: The transformation of the newly created picture which is the combination method of initial and transformed patterns by using the same method and same set of transformations.
Step 5: Repetition of the fourth step more and more as many times as possible i.e. theoretically infinite number of times. Some of commonly used IFS fractals are the Koch curve, Sierpinski Carpet, Sierpinski Gasket, Minkowski curve and the Koch Snowflake etc.

Fractal Dimensions
We know that the term Dimension can be used as the Topologic Dimension where a point is known for the dimension 0, a line is known for the dimension 1, a surface is known for the dimension 2 and a cube is known for dimension 3. So the dimension of an object can be represented by a number of parameters or coordinates [12]. Fractal dimension has a very unique feature. There are various types of representations used for the fractal dimensions like Hausdorf dimension, topological dimension, self-similarity Dimension and Box counting method dimension etc. And the self-similarity property is the major parameter which is used for the analysis and characterization of the different fractal geometries [13]. The self-similarity parameter of the fractal geometry is defined as D s = where s is the scaling factor and N is the number of selfsimilar structure copies. Fractal design has commonly two dimensions and components as A) The Initiator (0th stage) which is the basic structure of the fractal geometry. B) The Generator which is the structure which is repeated in a sequential pattern on the initiator in the consequent stages with the different dimensions. Fig.19: Fractal generator structure The Fractal generator can be used to unite or subtract the original area of the initiator n from each subsequent iteration stage in a regular pattern generating the required fractal structure.

Calculating Dimensions of Self-Similar Fractals
Generally linear fractals are used for calculation of the fractal geometry of the self-similar shapes. This is a mathematical approach and is limited to the mathematically predetermined fractal geometry structures producing precise values. [14]. So the dimension is calculated as D = .
Here the curve is enclosed and occupies a region of the restrained area as shown in the above figure.
The dragon curve exists in different forms with varying dimensions [15].

Lower dimensions, its area disappears
When for a basic dragon curve the line segments are reduced the fractal dimension of the seed is lowered and cannot cover an area. And it is done reducing the seed segments while the initial and finish points are kept constant simultaneously or by enhancing the distance between the start and finish points [16].
Varying of the dragon curves with suitable dimensions less than dimension 2 is shown in the following figure.

Calculating a Sierpinski pyramid (tetrahedron) dimension
Here it is studied the varying of dragon curve with component lengths i.e. the generator are enhanced related to its overall length.

Fig.23: Calculation of a Sierpinski pyramid or tetrahedron dimension.
Here the Sierpinski pyramid or tetrahedron contains a primary pyramid shaped structure that can be replaced by four smaller pyramids having half scaling of the original scale. So the calculated dimension is represented as D = .
Here the Sierpinski pyramid represents as a basic example of a two dimensional structure without any area. When such a Sierpinski pyramid exists that'll exist in a volume space i.e. its physical structure does not follow any other volume spaces [17].

Richardson's Dimension Equation
This method is used for calculation of fractal dimensions with the varying measurement lengths.
Here for the calculation of a dimension, first a establishment of a logarithmic relationship is required between the total measured length of the object and the length of the ruler used that is used to measure it. And this is done by taking graphs of the various values of Log p , where p is the length of the structure perimeter , against the corresponding values of Log ( ) , where s is called as scaling factor or the rulers length which is used for the perimeter measurement. Now the resulted slope of the graph is used to relate the fractal dimension and structure of the measured object. For a linear or smooth object, the smaller rulers are used to measure similar lengths similar to the larger ones, thus the slope is horizontal. And the graph with a horizontal slope used here corresponds to the dimension of 1. With the increase in the fractal dimension of any object, the level of the slope also increases [18].

Generation of some common fractal elements
The Fractal antenna is used to resolve the limitations of the conventional antenna like 1. The electrical size of the antenna affects the antenna performance i.e. affecting the antenna performance parameters like S 11 parameter, VSWR, gain, input and output impedance and radiation patterns etc. 2. Antenna size minimization is achieved by increasing the effective length of fractal antenna. a) Calculation of the Dimension of the Sierpinski Gasket This is a type of fractal geometry where a triangle is used as the basic geometry. The first iteration is taken by taking the midpoints of each side of an equilateral triangle and then connecting them to form another equilateral triangle which is slotted out from the original one. The higher iterations are achieved by repeating this same process into infinite number of times. Here the multiple iteration steps of Sierpinski gasket are shown in the following figure [19].

Fig.25: The Multiple iteration steps of Sierpinski Gasket
Also we can prove that the fractal dimension is not an integer by using Sierpinski Gasket fractal geometry as an example. Now we'll determine how the size of any object behaves with the increase in its linear dimension. And in the one dimensional case, we can take a line segment for the consideration. When the linear dimension of the line segment is doubled, then it can be observed that the characteristic size i.e. length of the line is also doubled [20].
In the two dimensional case, if the linear dimensions of any shape like a square is doubled then it can be observed that the characteristic size i.e. the area is increased by a factor of 4.
In the three dimensional case, if the linear dimension of any shape like a box is doubled then it can be observed that the characteristic size i.e. the volume is increased by a factor of 8. This relationship between linear scaling L, the dimension D, and the result of size increasing S can be represented as S = L. D Rearranging the above formula and taking logarithm we'll obtain as D = The above equation shows an expression for the dimension which is used to change the size as a function of the linear scaling.
Here the value of D is any integer like 1, 2, 3 …..n and it depends on the dimension of the geometry. And this above relationship holds good for all the Euclidean geometries and shapes. But now the question is how about fractals? But a different concept is used for fractals geometries and shapes. From the above figure it is clear that in the first step of building the Sierpinski gasket, if the linear dimension of the basic geometry triangle ( L) is doubled, then it can be observed that the area of total fractal geometry increases by a factor of 3 ( S). Now by using this pattern as stated above, the dimension for fractals geometry based on the Sierpinski gasket is calculated as D = = 1.585 And this calculation result proves regarding non-integer fractal dimension.

b) Calculation of the Dimension of the Koch Curve
Now we can use the formula which is derived earlier to generate the Koch curve. The construction of this curve is uniform and we can calculate its dimension can be calculated easily comparative to other dimensions.

c) Calculation of the Dimension of the Koch Snowflake
The fractal dimension based on Koch Snowflake can be constructed by using triangle as the basic curve. We can start the construction by using an equilateral triangle with each side of length L. A new triangle is created by connecting the middle of each side and the new triangle is of one-third size of that of the original. The same process is repeated for an infinite number of iterations. So here the length of the boundary is 3.
. But the area becomes less than the area of a circle drawn around the original triangle i.e. an infinitely long line surrounds a finite area. And the construction of a Koch Snowflake looks very similar to the coastline of a sea-shoreline as shown in the following figure [21]. Sierpinski. Here we can start the construction by using an equilateral triangle as the patch with sides of any length L. The middle point of each side is taken and connected them to make a new triangle and this new triangle is taken out of the triangle shaped patch. This step is called as the first iteration. And the same process is repeated for the multiple numbers of iterations [22].  [23] (1996) proposed a fractal monopole antenna that is designed using fractal geometry. Here the authors suggested that by using the Sierpinski gasket fractal geometry that follows the self-similarity property enables the proposed antenna to behave as the multi-resonating antenna. Here the proposed antenna is designed, simulated and validated using FDTD technique and it resonates up to five bands self-similarity property of the antenna. C. Punete et al. [24] (1998) presented a Microstrip patch antenna based on the Sierpinski fractal geometry suitable for multiband applications and compared it with a single band bow-tie antenna. The proposed antenna showing selfsimilarity properties is simulated and fabricated. And the author concluded that better performance parameters like return loss, VSWR, multiband operation and radiation patterns are reflected by the experimental and simulated results. The designed antenna exhibits a multi-frequency behavior and an efficient current distribution, but changes at all resonant modes. This efficient current distribution change provides various radiation characteristics resonating at higher frequencies showing radiation pattern with increased number of grating lobes. Here antenna element is designed on the dielectric laminate area that is extended beyond the circuitry of the device.
Here authors have suggested that the proposed antenna is operational at frequency range that can be used for allocation of 2.4 / 5.2 GHz ISM bands using only a single microstrip feed and also without using any matching network.
Authors have also presented different modification techniques allowing different bands allocation without using similarity factor i.e without changing flare angle of triangles. Here the authors have suggested that the proposed antenna uses two numbers of substrate layer of FR4 material with dielectric constant of ε r =4.4. Here these two substrate layers are of thickness of 1.6mm and are placed vertically one over another. Here the authors have suggested that due to these two stage substrate layers this proposed antenna behaves as multiband antenna. Abdelati Reha et. al. [36] (2018) presented a CPW fed fractal antenna based on sierpinski geometry. Here the authors suggest that the inter-relationship between the number of iterations and the resonant frequencies i.e. when the number of iteration increases the resonant frequency decreases with a constant ratio. Here the relative permittivity of the substrate material is increased; the proposed antenna shows improved performances. From the above survey, it is clear that the Sierpinski gasket geometry maintains symmetry, so provides multiband behavior. It shows an efficient current distribution, thus improving radiation patterns. It also provides large broadband performance, so useful in broadband antenna design.

Based on frequency independent fractal geometry
In this section, we have discussed on frequency independent antenna based on the fractal geometry. Here some major features of frequency independent antennas like LPDA antenna structures, antenna structures using EBG materials, reconfigurable antenna structures, defective ground structure antennas, and multi layered fractal structured antennas are discussed. From the above survey, it is observed that fractal geometries using self-similar property are used in frequency independent antennas suitable to transmit and receive over a large range of frequencies.
The frequency independent antennas shows steady performance over large frequency range provides condensed mutual coupling. These fractal geometries are also suitable for design of fractal array antennas.

Based on Minkowski island fractal geometry
In this section, we have discussed on fractal antenna structures using Minkowski fractal geometry. The Minkowski fractal geometry uses a generation curve as the basic structure and it is named according to the mathematician H. Minkowski. Here we can start the construction by using a square shape geometry in which each of four straight sides of square is being replaced with the generator curve and then iteration width is applied by a scaling of 1/3 to each straight sides of the square at the each iteration. Here the depth is adjusted to an optimization in order to achieve accurate result steps for construction of minkowski geometry as shown in figure [43]. Here the authors have used the multiport network method to validate the performance of the patch antenna. Here the Minkowski geometry is used to replace the patch side opposite to the position of feed arm for microstrip patch square ring antenna. Here by using the Minkowski geometry the Dual frequency band operation is achieved. Here the resonant characteristics are controlled by increasing the width of the sides. Authors have controlled that the impedance matrix (multiport network model) of the proposed antenna is analyzed and simplified using the selfsimilarity property of the fractal geometry with the greater precision and high accuracy with the less analysis time. From the above survey, it is achieved that this type of fractal geometry is suitable for broadband antenna designs in the field of wireless applications.

Based on Koch curve/island fractal geometry
In this section, we have discussed on fractal antenna structures using Koch curve fractal geometry. The fractal antenna based on the Koch Curve geometry is an antenna where a generation curve is used as the basic structure and it is named according to the mathematician H. V. Koch. Here we can start the construction by using a straight line and then this straight line is sub divided into three equal parts. And the middle segment is substituted with two other segments which are of approximately same length. It is the first iteration step of Koch Curve fractal antenna. And the same process is repeated for the multiple numbers of iterations [51]. Here authors have suggested that by using Koch snowflake structure an impedance bandwidth of 49 % can be achieved. Authors have also suggested that, a 70 % reduction in patch surface size in comparison to an ordinary Koch snowflake is achieved by using an air filled substrate and capacitive feed.   [59] (2018) presented a multiband non-planar steerable fractal dipole antenna based on Koch geometry. Here the proposed antenna uses fractal iterations, base angle and angle of steering through its parts providing very high directivity. From the above survey, it is clear that this is the most fundamental fractal geometry that provides improved performance parameters particularly good input impedance matching and high efficiency. Enclosure area of this fractal geometry is comparatively less than Hilbert curve geometry. Matching circuitry is also not required for Koch curve geometry.

Based on Hilbert curve fractal geometry
In this section, we have discussed on fractal antenna structures based on Hilbert curve fractal geometry. The fractal antenna based on the Hilbert Curve geometry is an antenna where a generation curve is used as the basic structure and it is named according to the mathematician D.
Hilbert. This fractal antenna uses the simplest geometry as it covers the area it occupies. And as here the line segments are not intersecting to each other, thus property reduces the complexity in the antenna structure [60]. Here two numbers of branches of radiating element are used to obtain dual-band performance. Tao Donga et. al. [65] (2018) presented a Electrically very small Hilbert fractal antenna. Here the authors have demonstrated that better impendence matching can be achieved by using near-field resonant parasitic (NFRP) antennas that uses Hilbert fractal geometry curves acting as the NFRP resonators and small vertical monopoles acting as the driven elements. The authors have suggested the advantages of this proposed NFRP antennas based on fractal geometry that there is no need of additional matching network and provides high radiation properties, while maintaining a small electrical size. From the above survey, it is observed that these types of fractal structures provide improved performance parameters particularly compact sized antenna structure. Hilbert curve provides a mapping algorithms, so it is largely used in the field of computer science.

Based on sierpinski carpet fractal geometry
In this section, we have discussed on fractal antenna structures based on sierpinski carpet fractal geometry. The fractal antenna based on the Sierpinski carpet geometry is an antenna where a rectangle is used as the basic structure and it is named according to the mathematician W. Sierpinski. Here the length, width and the total area are calculated using geometric progressions. We can start the construction by a rectangle as the patch with the dimension of any length and width (LХW). Then a new rectangle is created by considering a central slot in the patch. And the central slot size is considered as , where n is any integer. Here the same scaling is property is used in higher iterations also. And the same process is repeated for the multiple numbers of iterations. This is the simplest fractal antenna and it is most widely used fractal antenna as it follows multiband operations [66].  This modified antenna consists of a parasitic patch and an active patch. Authors suggested that when the parasitic patch is added, there is improve in performance parameters like the bandwidth (BW) is increased through a factor of 15 and it is compared to the single active element, a bandwidth (BW) = 2.7% which is at SWR=2:1 and the radiation parameter efficiency of 84%. S.N. Sachendra et al. [69] (2007) presented a sierpinski carpet fractal antenna using self-affine fractal geometry.
Here the proposed antenna is designed by applying scaling property to a square through a factor of 3 in the horizontal direction and through a factor of 2 in the vertical direction, providing nine rectangles and now out of these rectangles the central rectangle is removed. This process is called as the first iteration step. The same process is repeated for the remaining rectangles which are called as the second iteration step and the process can be continued upto infinite steps. This process is called as the iterated function system (IFS) and is a very useful tool in the fractal antenna designing. Here the authors have suggested that after the theoretical design procedure, CST EM software is used for the numerical simulation up to 2nd iteration to obtain design parameters like patch size and feeding location. Authors have suggested that when the number of iteration steps increases the antenna bandwidth increases and in the second iteration the antenna starts showing the multiband behavior. The proposed antenna shows improvement in different performance parameters like better return loss, low VSWR, large bandwidth, high directivity and gain. F. D. Nicola et. al. [75] (2018) presented a Multiband plasmonic fractal antenna based on sierpinski carpet geometry. The presented antenna is constructed by using deterministic fractal antennas. Here the antenna structure shows the enhanced localized electromagnetic fields in the infrared range with a hierarchial spatial distribution. The proposed antenna is a modified plasmonic multiband antenna based on metamaterial design which is based on the space-filling sierpinski carpet fractal with an independent tunable and polarization optical response exhibiting multiple resonances from the visible to mid-infrared rage. The proposed antenna uses gold SCS that is fabricated by electron beam lithography used on CAF2 and Si/Sio2 substrates allowing resonances originating from the diffraction mediated localized surface plasmons which can be tailored in the deterministic fashion by tuning the size, shape and position of the fractal elements. Authors have suggested that SCs with high order of complexity provides a strong and electromagnetic near-field of plasmonic modesthus distributed hierarchically and provides a technique for the realization of compact active devices with a broadband and strong spectral response operating in the visible/mid-infrared range. The proposed antenna provides adavantages of surface enhanced raman spectroscopy (SERs) used on the brilliant Cresyl Blue molecules deposited onto plasmonic SCs. The authors have concluded that the proposed antenna can achieve a broadband SERs enhancement factor up to 10 4 , thus can be used for the chemical diagnostics applications. From the above survey, it is observed that these types of fractal structures provides improved performance parameters particularly good impedance matching, impendence bandwidth, wideband/multiband, frequency independent, high radiation patterns. With the advantages of the condensed mutual coupling it is also suitable in design of fractal array antennas.

Based on hybrid fractal geometry
In this section, we have discussed on fractal antenna structures based on some hybrid fractal geometry. Hybrid fractal antennas are combinational structured antennas those use a combination method of different antenna structures. And it is an effective technique for improving the fractal antenna performance. The Hybrid fractal antennas are designed by using the following possible combinations [76].
-Sierpinski -Meander -Giuseppe peano There are some other combinations of fractal geometries as shown in the figure.  Here the presented antenna is combinational antenna which is the combination of fractal geometry and meander line. Here authors have demonstrated that the use of defected ground structure enables enhanced bandwidth in the microstrip patch antenna. The proposed antenna is of the dimension of (45×40×1.6) mm 3 , provides two resonance frequencies operating at 3.79GHZ and 5.5 GHz with wide band ranging in 1.31GHz to 6.81 GHz i.e. a bandwidth enhancement of 135% is achieved with high gain of 6.5 dBi. The authors have concluded that the proposed antenna is simulated and fabricated and it is found that simulated and experimental results are matching in good agreement with each other. From the above survey, it is observed that these types of fractal structures provide improved performance parameters particularly antenna minimization, high impendence bandwidth, high radiation efficiency, high gain, high radiation patterns and multiband/wideband properties.

V. DISCUSSION ON FEATURES OF FRACTAL GEOMETRY
Fractal geometries shows various major advantages, hence these are widely used now a days for different applications. Some major advantages, disadvantages and applications are listed as follows.

5.1
Advantages and disadvantages of using fractal geometry Advantages: I.
Smaller in size, Lighter in weight and less in cost II.
Use of fewer components III.
Provides space filling, self similarity and scaling properties IV.
Provides Increased bandwidth V.
Provides Enhanced gain and directivity VI.
Used for good input impedance matching VII.
Provides multi-frequency performance VIII.
Uses the essential combination of inductance and capacitance providing multiple resonances for the similar structures. IX.
Using fractal compression technique the image can be enlarged, so there is no need for pixelisation. Disadvantage: I.
Complexity in modeling the antenna II.
Loss of gain in some cases III.
Numerical analysis limitations IV.
Not so beneficial after first few iterations

Major applications of fractal geometry I.
Wideband antenna designs II.
In design of frequency selective surfaces(FSS) III.
Biological Sciences i.e to analyze biomedical phenomenon like bacteria growth pattern, pattern of nerve dendrites etc. IV.
In fluid mechanics V.
In computer graphics VI.
Analyze seismic patterns VII.
Fractal image rendering and image compression schemes

VI. CONCLUSION
Several fractal geometries and their applications to various antenna designs are properly presented. This expands the area of research in wireless communication. Most of the nature inspired and human inspired fractal geometries are briefly summarized. The role of the mathematical modeling, the implementation procedure corresponding to the mathematical expressions is successfully explained. Low profile, low cost, and light weight antennas being popular are also discussed considering various fractal geometries. Broadband, wideband, and multiband being the most desirable characteristics of all the antennas are properly discussed considering the implementation of fractal geometry. Advantages, disadvantages, major applications, and future scope are properly presented to highlight the importance of the fractal geometry.

VII. FUTURE SCOPE
I. The detailed study the biosensor attractions can be done using fractal geometry. II. Fractal geometry mesh generation can also be used to reduce memory size requirements and analysis of finite element for CPU of the vibration problems by using space filling nature of fractal geometries. III. In computer applications, the fractal image compression and Fractal image rendering schemes can also be used for the memory size reduction and processing time reduction. IV. Other modifications like bending or folding the antenna, introducing fractal geometry of the radiator, variation of length of ground plane, varying substrate thickness can be considered to study effects on the radiation properties.
V. Using fractal geometries 3D polymer printing and metallization could be an alternative and effective method for low-cost rapid prototyping and testing of new and complex antenna systems.