# Fractal Metasurfaces and Antennas: An Overview for Advanced Applications in Wireless Communications
## 2 Fractal Antenna Elements  
The general concept of fractal can be applied to design exotic antenna elements, offering improved and exciting levels of performance with respect to those of conventional antennas. Fractal antenna elements may have a compact size, as well as multiband and/or broadband capabilities. The design of fractal antennas involves the reiteration of a specific geometric pattern at different scales within the antenna structure. This self-similar pattern allows us to increase the overall antenna length within a limited physical space. Furthermore, fractal radiators can achieve multiple frequency bands because different parts of the antenna are similar to each other at different scales. As a result, fractal antenna elements can be compact and multiband, offering broader frequency coverage as compared to that of traditional antennas. The concept of fractal antennas dates back to the late 1980s. In 1986, [PERSON] and [PERSON] demonstrated the use of fractal self-similarity in antenna array synthesis [18]. In 1988, the physicist [PERSON] built the world's first fractal element antenna. He proposed that, by incorporating self-similar geometric shapes into the antenna's structure, it is possible to improve its performance and miniaturization skills [19; 20]. In 1995, [PERSON] co-founded Fractal Antenna Systems Inc. to begin delivering the world's first fractal-based commercial antenna solutions [21]. In the mid-1990s, [PERSON] et al. [22] demonstrated the multiband capability of fractals by exploiting the Sierpinski monopole and dipole behavior. From the works of [PERSON] and [PERSON] onwards to the present day, the inherent advantages offered by fractal antennas have attracted considerable interest among researchers and engineers in the telecommunications field, leading to further exploration and development of fractal antenna technology. Nowadays, fractal antennas are used in wireless communications systems, including mobile phones, Wi-Fi routers, and satellite communications [8; 9; 10; 11; 12]. Indeed, their compact size, multiband operation, and efficiency make them suitable for a wide range of wireless devices and networks. The figures below show some fractal antennas based on well-known fractal shapes, which are just a few examples of the various configurations discussed in the literature [23; 24; 25; 26; 27; 28]. In particular, Figure 2a illustrates the Sierpinski monopole introduced by [PERSON] in [26], which is able to offer multiband behavior. The Sierpinski triangle is formed by subtracting a central inverted triangle from a main triangle shape, as demonstrated in Figures 1b and 2a. This process leaves three equal triangles on the structure, each with sides half the size of the starting triangle (see Figure 1b). The same subtraction procedure can be iterated for the remaining triangles. Thus, each one of its three main parts is exactly equal to the whole object, but scaled by a factor of two, and so on. Due to these particular similarity properties, the Sierpinski gasket can offer multiple frequency bands because different parts of the antenna are similar to each other at different scales. As illustrated in Figure 2, the Sierpinski monopole proposed in [26] offers penta-band behavior. It can be seen that each of the five subgaskets in Figure 2a (circled structures) is a scaled version of the whole structure, thus offering five different operating bands, as shown by the input reflection coefficient in Figure 2b. The figure also shows the parameters relative to each band, including the frequency values \(f_{n}\) (\(1\leq n\leq 5\)) at the minima of the measured input return loss (\(L_{r}\)), the relative bandwidth (_Bwidth_), and the frequency ratio between two adjacent bands (\(\delta=f_{n+1}\)/\(f_{n}\)). Figure 2b demonstrates that the bands of the Sierpinski monopole are log-periodically spaced by a factor of two (\(\delta\cong 2\)), which corresponds to the same scale factor existing among similar structures on the fractal shape.