Collatz's Ant and Similarity of Landscapes

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. The question is: what makes a certain landscape specific? As already seen before in the previous post, the answer isn’t the stopping time. In fact, there are vastly different landscapes with the same stopping time. One can also wonder if the maximum distance ... Read full article.