Collatz's Ant and Landscape Similarity: The Mystery of Beta

This post explores the similarity of landscapes generated by Collatz's ant trajectories. By analyzing stopping time (τ), maximum Euclidean distance (α), the step at which the maximum distance is reached (β), and the final distance (γ), the author finds that stopping time is not a decisive factor in landscape similarity. While maximum distance (α) is related to landscape scale, it's insufficient to distinguish different landscapes. However, the step at which the maximum distance is reached (β) appears to be an indicator for distinguishing different landscapes, but the underlying mechanism requires further investigation. The article presents multiple examples showing the complex relationship between β and landscape shape and poses some unsolved mysteries, such as why, when the maximum distance (α) is different, β is sometimes the same and sometimes different? This provides a new perspective on the study of the Collatz conjecture.