Hatami Configurations: A Computational-Geometry Case Study on Collatz Trajectories

We introduce a novel geometric framework for analyzing discrete dynamical sequences by embedding their characteristic points into the Euclidean plane and extracting a consistent set of geometric descriptors. As a case study, we apply this framework to Collatz trajectories, defining what we call the Hatami Configurations and classifying them into distinct geometric types. Using segment lengths, orientation angles, direct distances, and divergence measures, we provide a systematic methodology for geometric classification and statistical evaluation. Our exhaustive computations for every integer 2≤n≤30,000 reveal clear structural patterns, significant differences between prime and composite starting values, and stable scaling relations between initial values and direct distances. Although demonstrated here on the Collatz problem, the methodology illustrates how computational geometry can serve as a general tool to explore number-theoretic and dynamical problems.