## 2 Methodology
### The Eisenstein Fractal Generator  
The Eisenstein fractal was previously introduced in [23], where it is represented as a subset of Eisenstein integers; this set is denominated as the Eisenstein domain, which is characterized by its hexagonal lattice grid. Eisenstein integers are complex numbers of the form \(z=a+b\omega\), where \(a,b\) are real integers and \(\omega\) is a primitive cube root of unity given by \(\omega=e^{2\pi i/3}\). The Eisenstein fractal at a given growth stage \(p\) is represented by the set of points \(S_{p}\). The initial stage (\(p=1\)) is defined by the vertex set \(v=\omega^{2},\omega^{1},\omega^{0},0\), as illustrated in Figure 1, which establishes the base geometry of the fractal generator. For higher stages (\(p>1\)), the generator is constructed by replicating the previous stage's point set at each vertex, incorporating a rotation pattern \(r_{p}=e^{(p-1)\pi i/3}\) (\(60^{\circ}\)) and a scaling factor \(\delta_{p}=2^{p-1}\) corresponding to the growth level. The points for the subsequent stage are recursively defined in terms of the previous stage as follows:  
\[S_{p+1}=\bigcup_{v}S_{p}+(\delta_{p}\cdot r_{p}\cdot v) \tag{1}\]  
The equation for the generator at any growth stage is described as  
\[S_{p}=\bigcup_{v}\left(\sum_{p=1}^{p}2^{(p-1)}\cdot e^{(p-1)\pi i/3}\cdot v\right) \tag{2}\]The Eisenstein fractal evolves through recursive geometric transformations and is described by the point set \(S_{p}\). In Equation (1), the operator \(\bigcup_{v}\) denotes a union over all base vertices \(v\in\{\omega^{2},\omega^{1},\omega^{0},0\}\). The term \(S_{p}+(\delta_{p}\cdot r_{p}\cdot v)\) represents a translation of the entire point set \(S_{p}\), where the translation vector is computed by scaling and rotating the vertex \(v\). Specifically, the scaling factor \(\delta_{p}\) controls the growth size at stage \(p\), and the rotation factor \(r_{p}\) applies a complex exponential rotation by \(60^{\circ}\) increments, preserving the hexagonal symmetry of the Eisenstein lattice. Equation (2) expresses this recursive process in a closed form. Here, the inner summation sequentially applies the same scale and rotation operations used in the recursive definition, accumulating the transformation effects across all stages from \(1\) to \(P\). The outer union \(\bigcup_{v}\) again distributes these transformations across each vertex \(v\), generating the complete point set \(S_{P}\). The recursive geometric construction of the Eisenstein fractal across successive growth stages is represented in Figure 2, where each new stage replicates the previous stage's structure around rotated and scaled versions of the Eisenstein integer vertices. Table 1 lists the complex coordinates \(a+jb\) of the Eisenstein integers used at stage \(p=2\). Each entry in the table corresponds to a specific vertex location generated by linear combinations of scaled roots \(\omega^{k}\), revealing the structured placement of elements in the generator. The self-similar nature of fractal geometry allows each iteration of a structure to be systematically derived from its preceding stage, streamlining element distribution as the array scales to larger configurations. In antenna array design, this property facilitates the development of large-scale arrays that retain geometric uniformity and modularity. In this study, we leverage the inherent self-similarity of Eisenstein fractals in conjunction with array thinning techniques to maintain wideband performance while significantly reducing the number of active elements. To construct thinned antenna arrays across multiple fractal growth stages, we introduce two structural transformation mechanisms based on the underlying fractal generator. Figure 1: Eisenstein domain lattice showing first Eisenstein integers \(0\), \(\omega^{0}\), \(\omega^{1}\), \(\omega^{2}\). Figure 2: Geometric construction of the Eisenstein fractal array across growth stages, showing recursive replication at rotated vertices for stages: (**a**) \(p=1\), (**b**) \(p=2\), (**c**) \(p=3\), (**d**) \(p=4\).