Ants on the highway

TL;DR

This paper explores Generalised Langton's Ants, discovering rules with numerous highways, analyzing their structure, and proving that the number of possible highways can be infinite, with some appearing extremely infrequently in simulations.

Contribution

It introduces new rules with many highways, provides structural analysis, and proves the potential for infinitely many highways in generalized Langton's Ants.

Findings

Rules with many highways were discovered.

The number of highways can be infinite.

Highway frequencies vary greatly, with some being extremely rare.

Abstract

We perform intensive computations of Generalised Langton's Ants, discovering rules with a big number of highways. We depict the structure of some of them, formally proving that the number of highways which are possible for a given rule does not need to be bounded, moreover it can be infinite. The frequency of appearing of these highways is very unequal within a given generalised ant rule, in some cases these frequencies where found in a ratio of $1/10^7$ in simulations, suggesting that those highways that appears as the only possible asymptotic behaviour of some rules, might be accompanied by a big family of very infrequent ones.