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Collatz's Ant and Similarity of Landscapes

This is a small development from the previous post, which is mostly focused on trying to understand where the similarities between landscapes come from (and what can be taken as proxies for these). Considering the trajectories respective to the numbers from $n = 10^{20}$ to $n = 10^{20} + 100$, and the corresponding stopping times ($\tau$), maximum euclidean distance hit ($\alpha$) from the origin point $(0, 0)$ where the ant starts, the step at which such maximum distance is hit ($\beta$), and also the distance from the origin at the last step ($\gamma$) (in the third plot it’s normalized by $\alpha$), we have the following: along with the corresponding landscapes for each trajectory.

This is a small development from the previous post, which is mostly focused on trying to understand where the similarities between landscapes come from (and what can be taken as proxies for these). In fact, if we think about scaling up the same pattern such that $\alpha$ increases, we’re more or less talking about the same landscape but with vastly different dimensions (such that there’s some type of scale-free similarity). One thing that’s easily noticeable from the start is that of $\beta$ or the step at which the maximum distance is reached.

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