# Hexagonal Game of Life

Inspired by this post on reddit, I decided to investigate cellular automata which use the hexagonal tesselation of the plane, and have found a rule in which supports a spaceship. I’ve made a web-based simulator which I’ve been using to investigate these automata.

I first investigated a lot of Game of Life style rules, where the status of a cell in the next iteration depends only on its current status and the number of live neighbors. I failed to find any such rules which were particularly interesting (although there does exist at least one such rule which supports a spaceship).

I then looked at a more general kind of rule which is no longer invariant under any permutations of the neighbors, but only under rotations and reflections. There are 13 possible neighborhoods up to these symmetries. I have labelled them as 0-6 and a-f, as shown here:

The first rule I’ve found supporting a spaceship is 2b/2ab (that is, a live cell remains alive if its live neighbors have the patterns 2 or b, and a a dead cell becomes alive when it has live neighbors in the patterns 2, a, or b). This is essentially like a conventional S2/B2 rule, except that one S case is removed. This extra chance for cells to die prevents the patterns from exploding.

The spaceship has period 3, and looks like this:

The following very simple patterns are predecessors of the spaceship. The left one just creates a spaceship, the right one creates two spaceships traveling in opposite directions:

I have found some oscillators as well, many of which are in common with the linked reddit post:

The following pattern is a predecessor for an amazing period-126 oscillator, which rotates itself 1/6 of the way through a circle every 21 generations:

The first pattern I’ve found in this rule that grows infinitely is a modified version of the spaceship which creates a trail of oscillators. It looks like this:

I have yet to discover any guns, but I have only started to investigate this rule, and there are many other such rules which support spaceships as well.