Collatz's Ant and Similarity of Landscapes

Summary

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 hit would give a clue. I have no intent of going about labelling each landscape according to which potential category it might belong to, so as to find if there are any discrepancies to be pointed at, but from a first look at such landscapes and also relying trivially on our intuition, we can of course state that $\alpha$ is always going to be minimally associated with different landscapes to the extent that these can be purely differentiated from each other given their dimensions. Beyond this, it’s not useful. 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). Not that there are any examples of that in these trajectories, as that would also presumably only be noticeable when comparing trajectories with $\tau$’s with different orders of magnitude. But this suffices to make the point: $\alpha$ alone isn’t sufficie...