This paper presents an analytical study of the use of mathematical modelling, such as the iterated function system (IFS), for patch fractal antenna design. Current, modern research in the field of communications focuses on the realization of compact communication systems. Therefore, they concentrate on reducing the size of individual devices and the constitution of elements, particularly antennas. Fractal geometries are generated by recursive transformations, which provide a rigorous mathematical framework for constructing self-similar sets through affine contraction mappings using an iterative procedure for compact, miniaturized self-similar structures. With each iteration n, the effective electrical length of the antenna increases. Finite element simulations validate the theoretical predictions, demonstrating close agreement between calculated and simulated resonant frequencies so that additional resonant frequencies appear, which enable wideband and multiband operations. The results confirm that the iteration order can be used as a systematic design parameter to achieve target frequency bands without relying just on numerical optimization. This work illustrates how mathematical concepts such as IFS and recursion can guide the analytical design of compact, multiband, or wideband fractal antennas for wireless communication systems and aims to demonstrate the role of applied mathematics in the systematic design and miniaturization of fractal antenna structures.