On the Determination of Centers of Mass via Fractal Calculus and Its Applications in Board Games

This study introduces a novel approach to chess analysis based on center-of-mass dynamics and discrete fractal derivatives, offering an alternative framework for evaluating gameplay strategies. Unlike conventional methods that rely on exhaustive search and statistical simulations, our model provides a macroscopic perspective by analyzing the collective motion of pieces over time. By representing chess positions as a dynamic system in R2, we identify key movement patterns—such as oblique, parallel, and orthogonal trends—revealing strategic tendencies throughout the game. Additionally, fractal derivatives enable the detection of subtle momentum shifts and long-term imbalances, enhancing the understanding of decision-making processes. This approach is computationally efficient and adaptable, extending beyond chess to applications in sports analytics and real-time strategy games. These findings highlight the potential of interdisciplinary techniques in capturing complex strategic behavior within dynamic environments.