# Fractal Antennas: An Historical Perspective

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. About the Fractal Geometry

#### 2.1. Recursive Relation Based

^{2}+ c

#### 2.2. Iterated Function System (IFS)

#### 2.3. L-Systems

#### 2.4. Random

## 3. Fractals in Antenna Engineering

#### 3.1. Mandelbrot

#### 3.2. Sierpiński

_{i}is the affine linear transformations [9]. This affine transformation is defined as:

_{0}and y

_{0}are translations.

_{1}, the entire structure is responsible for the radiation. Therefore, if the height of this monopole is designed to have a length of λ/4, the radiation pattern is obtained with a zero in the zenith direction and with maximum radiation on the horizon (in case of conductors of finite conductivity and finite size, the maximum rises above the horizon. If for a second frequency band with central frequency in f

_{2}(f

_{2}= 2 · f

_{1}) it is achieved that the height triangle h

_{2}(h

_{2}= h

_{1}/2) practically contributes to the radiation, a type radiation pattern will be again as a λ/4 monopole. Additionally, if it is repeated for the next iterations, the same conclusion is reached. In this way, quite similar radiation patterns can be achieved in various frequency regions in addition to a good adaptation for each of the regions (Figure 8 and Figure 9).

_{30}), but with large secondary lobes. Therefore, not only fractal inspired boundaries but also mass-fractals are useful to obtain microstrip patch antennas with higher-order modes featuring a larger directivity than the fundamental mode and at the same time keeping a broadside radiation.

#### 3.3. Koch

#### 3.4. Hilbert

#### 3.5. Cantor

_{0}to K

_{3}, the antenna started covering the lower bands, which include personal and commercial communication applications.

#### 3.6. Minkowski

#### 3.7. Peano

## 4. Evolution of Publications in the Fractal Antenna Field

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Benoit Mandelbrot (1924–2010), polish mathematician, considered as the father of fractal geometry.

**Figure 2.**A Mandelbrot based microstrip patch antenna generated using z

_{n+1}= z

_{n}

^{4}+ z

_{0}

^{2}− 1 [12].

**Figure 3.**Simulated current distribution at f = 2.75 GHz using a Method of Moments code. The black arrows (intentionally added) indicate the main flow of the current on the antenna surface.

**Figure 5.**Waclaw Sierpiński (1882–1969), polish mathematician. Well-known fractals such as the Sierpiński triangle or carpet are named after his contributions.

**Figure 7.**A monopole antenna inspired in the Sierpiński triangle [13].

**Figure 9.**Measured radiation cuts at four frequency bands of the Sierpiński antenna of Figure 7.

**Figure 10.**A Sierpiński based microstrip patch antenna fed with a coaxial probe. The size of the ground plane is the same size of the substrate.

**Figure 11.**Measured radiation patterns for the fundamental mode and a high-directive mode of the Sierpiński antenna of Figure 10.

**Figure 12.**Niels Fabian Helge von Koch (1870–1924), Swedish mathematician. The Kock snowflake is one of the most well-known fractals.

**Figure 13.**A microstrip patch antenna based on Koch island printed on a foam substrate presenting very low weight.

**Figure 16.**David Hilbert (1862–1943), German mathematician. The Hilbert curve has been used to design small antennas for wireless applications due to its space-filling properties.

**Figure 17.**Several iterations of the Hilbert curve to generate a monopole: From the straight monopole (7 cm) to the 5th iteration. Feed point is shown at the base. All antennas are mounted vertically on a ground plane 80 cm × 80 cm [62].

**Figure 18.**A very small antenna inspired in the Hilbert curve. The antenna operates a WiFi 2.5 GHz and 5 GHz bands within a reduced space of 7 mm × 3 mm × 2 mm. A = 55.4, B = 44.2, C = 9.2, D = 6.0, E = 0.0, F = 2.4, G = 14.3, all cotes in mm. Clearance area is D × G.

**Figure 19.**Measured VSWR (Voltage Standing Wave Ratio) and total efficiency for the antenna shown in Figure 18.

**Figure 21.**Cantor geometry-based antenna is designed for RF energy harvesting [87].

**Table 1.**Affine linear transformations to generate the Koch island of Figure 13.

IFS to Generate the Koch Island Based on Seven Affine Linear Transformations | |
---|---|

$\begin{array}{l}{v}_{1}\left(x,y\right)=\left(\begin{array}{cc}{\scriptscriptstyle \frac{1}{2}}& {\scriptscriptstyle \frac{-\sqrt{3}}{6}}\\ {\scriptscriptstyle \frac{\sqrt{3}}{6}}& {\scriptscriptstyle \frac{1}{2}}\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\\ {v}_{2}\left(x,y\right)=\left(\begin{array}{cc}{\scriptscriptstyle \frac{1}{3}}& 0\\ 0& {\scriptscriptstyle \frac{1}{3}}\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)+\left(\begin{array}{c}{\scriptscriptstyle \frac{1}{\sqrt{3}}}\\ {\scriptscriptstyle \frac{1}{3}}\end{array}\right)\\ {v}_{3}\left(x,y\right)=\left(\begin{array}{cc}{\scriptscriptstyle \frac{1}{3}}& 0\\ 0& {\scriptscriptstyle \frac{1}{3}}\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)+\left(\begin{array}{c}0\\ {\scriptscriptstyle \frac{2}{3}}\end{array}\right)\\ {v}_{4}\left(x,y\right)=\left(\begin{array}{cc}{\scriptscriptstyle \frac{1}{3}}& 0\\ 0& {\scriptscriptstyle \frac{1}{3}}\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)+\left(\begin{array}{c}{\scriptscriptstyle \frac{-1}{\sqrt{3}}}\\ {\scriptscriptstyle \frac{1}{3}}\end{array}\right)\\ {v}_{5}\left(x,y\right)=\left(\begin{array}{cc}{\scriptscriptstyle \frac{1}{3}}& 0\\ 0& {\scriptscriptstyle \frac{1}{3}}\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)+\left(\begin{array}{c}{\scriptscriptstyle \frac{-1}{\sqrt{3}}}\\ {\scriptscriptstyle \frac{-1}{3}}\end{array}\right)\\ {v}_{6}\left(x,y\right)=\left(\begin{array}{cc}{\scriptscriptstyle \frac{1}{3}}& 0\\ 0& {\scriptscriptstyle \frac{1}{3}}\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)+\left(\begin{array}{c}0\\ {\scriptscriptstyle \frac{-2}{3}}\end{array}\right)\\ {v}_{7}\left(x,y\right)=\left(\begin{array}{cc}{\scriptscriptstyle \frac{1}{3}}& 0\\ 0& {\scriptscriptstyle \frac{1}{3}}\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)+\left(\begin{array}{c}{\scriptscriptstyle \frac{1}{\sqrt{3}}}\\ {\scriptscriptstyle \frac{-1}{3}}\end{array}\right)\end{array}$ |

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**MDPI and ACS Style**

Anguera, J.; Andújar, A.; Jayasinghe, J.; Chakravarthy, V.V.S.S.S.; Chowdary, P.S.R.; Pijoan, J.L.; Ali, T.; Cattani, C.
Fractal Antennas: An Historical Perspective. *Fractal Fract.* **2020**, *4*, 3.
https://doi.org/10.3390/fractalfract4010003

**AMA Style**

Anguera J, Andújar A, Jayasinghe J, Chakravarthy VVSSS, Chowdary PSR, Pijoan JL, Ali T, Cattani C.
Fractal Antennas: An Historical Perspective. *Fractal and Fractional*. 2020; 4(1):3.
https://doi.org/10.3390/fractalfract4010003

**Chicago/Turabian Style**

Anguera, Jaume, Aurora Andújar, Jeevani Jayasinghe, V. V. S. S. Sameer Chakravarthy, P. S. R. Chowdary, Joan L. Pijoan, Tanweer Ali, and Carlo Cattani.
2020. "Fractal Antennas: An Historical Perspective" *Fractal and Fractional* 4, no. 1: 3.
https://doi.org/10.3390/fractalfract4010003