Understanding and generalizing Langton’s ant

The ant moves infinite number of times, step by step. May turn either left or right. Arbitrarily  it turns left by default (albeit, as expected, it does not make any difference).When stands somewhere on  the grid, flips the current color of the grid element, so that it knows it had been there already. When  finds the place that had been visited earlier, turns the other way round, because needs to look for  new places.

The system related to the ant’s movement is likely to have a specific attractor, as in ca. 10k steps the behaviour of quasi-chaotic (ie. not chaotic at all, just looks chaotic) turns into the inifnite one of a very regular pattern, called the “highway”.

The very easy rules: change the direction if you at least once visited the place. The generalization would be  to remember all places the ant has visited (currently we remember in a binary way, which allows re-unvisiting the grid element when visited twice) with the next primes (allowing infinite number of memory for new places) and allowing making other than binary decisions (right-left) via the use of prime numbers (currently inifnite  amount of decisions would be possible).

Re-thinking the ant’s behaviour, we could notice that in every moment of time (defined as step), the ant  makes a binary decision based on somewhat crippled memory of earlier visited places. In reality, the perfect ant should remember all the visited places (grid elements in the lattice). Generalizing the binary decision into the one represented by an infinite set will be ommited later.

Thus, a binary decision and good memory of places. Still, in case the ant has a perfect memory of the places,  I have not been able to grasp the notion of the graphical representation of the primes on the grid (other than obvious ones), thus the very first research would be to put them in spiral way, which I will show later. The  ant will turn either left or right.

Concluding, two elements for research:

– formula for movement generalization for the non-binary decision non-crippled memory ant,
– code and graphs for binary decision non-crippled memory ant.

I will probably start with the second one and as soon as I have have both the code and graphs, I will do post them in the blog. I understood something about poker which I should not use against people. Thus I will also post it and will not take advantage of this approach.

assume