Scalable Complexity: Mathematical Analysis and Potential for Physical Applications of the Keçeci Circle Fractal

Mehmet Keçeci¹

¹ORCID : https://orcid.org/0000-0001-9937-9839, İstanbul, Türkiye

Received: 13.05.2025

Abstract:

 Many systems encountered in nature and engineering exhibit complex and hierarchical geometric structures. Fractal geometry provides a powerful tool for understanding and modeling these structures. However, existing deterministic circle packing fractals, such as the Apollonian gasket, often adhere to fixed geometric rules and may fall short in accurately reflecting the diversity of observed structures. Addressing the need for greater flexibility in modeling physical and mathematical systems, this paper introduces the Keçeci Circle Fractal (KCF, Keçeci Fractals (first defined: March 2025)), a novel deterministic fractal. The KCF is generated through a recursive algorithm where a parent circle contains child circles scaled down by a specific scale_factor and whose number (initial_children, recursive_children) is controllable. These parameters allow for the tuning of the fractal's morphological characteristics (e.g., density, void distribution, boundary complexity) over a wide range. The primary advantage of the KCF lies in its tuneable geometry, enabling more realistic modeling of diverse systems with varying structural parameters, such as porous media (for fluid flow simulations), granular material packings, foam structures, or potentially biological aggregations. Furthermore, the controllable structure of the KCF provides an ideal testbed for investigating structure-dependent physical phenomena like wave scattering, heat transfer, or electrical conductivity. Mathematically, it offers opportunities to study variations in fractal dimension and packing efficiency for different parameter values. In conclusion, the Keçeci Circle Fractal emerges as a valuable and versatile tool for generating geometries with controlled complexity and investigating structure-property relationships across multidisciplinary fields.

Keywords:

Fractal, Circle Packing, Recursive Algorithm, Deterministic Fractal, Keçeci Fractal, Keçeci Fraktal, kececifractals, Geometric Modeling, Tuneable Geometry, Porous Media, Self-Similarity, Parameter Space.