## 1 Introduction  
Fifth-generation (5G) communication has brought substantial enhancements to telecommunication systems, offering higher transmission rates and more efficient spectrum utilization. However, with the continuous evolution of technologies and the impending arrival of sixth-generation (6G) networks, there is a growing need for more advanced and efficient solutions [1]. In this context, the design and optimization of antenna array systems play a critical role in maximizing physical resource utilization and enabling greater control over signal directionality, noise mitigation, and other interferences. These advancements support practical applications such as ultra-reliable low-latency communication (URLLC), large-scale Internet of Things (IoT) deployments, sensor networks, and radar systems [2, 3]. Emerging technologies such as multiple-input multiple-output (MIMO), reconfigurable intelligent surfaces (RISs), and edge computing have facilitated the development of more complex antenna arrays with an increased number of elements. Simultaneously, improvements in computational power and optimization algorithms have expanded the possibilities for antenna design, enabling the exploration of novel geometries. Among these, fractal geometries have gained significant attention for their ability to support broader bandwidths and higher element densities [4]. The application of fractal geometry in antenna array design has seen rapid growth, driven by the intrinsic advantages that these structures offer. Notably, fractal-based arrays exhibit enhanced multiband operation, scalability for large-element configurations, and potential for miniaturization [5]. These characteristics are particularly advantageous for technologies such as MIMO and wideband, which have shown notable performance improvements in data transmission and spectral efficiency. The advancement of computational methods and design tools has further accelerated research in this area, enabling the investigation of innovative fractal structures and optimization strategies, and facilitating the design of increasingly complex and efficient arrays [6]. The incorporation of fractal geometry into antenna design began in the 1990s due to the favorable relationship between self-similar structures and frequency-independent behavior, which directly supports wider bandwidths [7]. These properties are preserved when fractal geometry is applied to the spatial distribution of array elements, leading to the development of fractal antenna arrays. Such arrays have demonstrated strong performance and are considered a viable alternative to conventional designs, offering improvements in both bandwidth and spatial efficiency [5,8]. Early studies such as [9, 10, 11, 12] evaluated the effectiveness of well-known fractals previously applied to individual antenna elements, including the Gosper curve, Fudgeflake, Cantor set, and Sierpinski fractals in the design of antenna arrays. Subsequent research introduced optimization algorithms to further improve array performance. For instance, ref. [13] proposed a broadband array design based on the space-filling Peano-Gosper curve, optimizing the element distribution to enhance bandwidth across different curve iterations. Similarly, ref. [14] employed self-similar geometries with rotational symmetry to suppress grating lobes in ultra-wideband (UWB) arrays, using a covariance matrix adaptation evolutionary strategy (CMA-ES). More recent studies, such as [15,16], introduced innovative methodologies for designing fractal antenna arrays. These approaches achieved high performance without increasing structural complexity, utilizing self-similar geometry-based fractal generators in heptagonal, hexagonal, and pentagonal configurations. Circular and spiral self-similar concentric ring arrays were explored in [17,18], where differential evolution (DE) and particle swarm optimization (PSO) algorithms were applied to optimize element distribution.