Exploring the nexus between thermodynamic phase transitions and geometric fractals through systematic lattice point classification

Fractals are ubiquitous in the natural world, and their connection with phase transitions has been widely observed. This study investigates mechanisms of fractal formation from the perspective of phase transitions. A novel set of probability calculation methods is introduced to establish a direct link between fractals and phase transitions. Notably, in the Ising model, a specific category of boundary lattice points undergoes a phase transition when the associated weight reaches ∼0.4. The identified correlation between phase transitions and fractals suggests the emergence of fractal structures at this critical weight. This paper offers supporting evidence for this conclusion through the deliberate manipulation of the proposed probability-based method. This research contributes to a deeper understanding of the interplay between fractals and phase transitions, providing valuable insights for further exploration in diverse scientific domains.