Math Burst is an annual 2-week math research project run in December that my school facilitates. The school puts students into groups which look into a particlar subject based off of interest. My group, advised by Dr Jonathan Quartin was tasked with looking into Cellular Automata, specifically a variation on John Conway's Game of Life, which we called Conway's Farming Simulator. Rules for the game are here.
For our presentation, we explored two different types of boards: finite boards and infinite boards. Since the main difference between the normal game of life and this is the inclusion of Plants, our questions focused on them and how they work in the game of life.
On finite boards, we asked two questions; how many Plants can be made, and how many generations we can go through before generating a Plant. On infinite boards, we noticed that some checkerboard patterns would expand indefinitely and others would stop, and we asked why they did that and which ones did which. We also looked at why patterns could expand infinitely. Documents on our findings have been linked appropriately.
I looked into general checkerboard (of water and plant cells) behavior, and checkerboards that grow infinitely and ones that don't..
Additionally, I found a shape made of grass that provably grows forever, and is emergent in many checkboards that grow infinitely, and helps to prove that they do. In some sense, it builds an infinitely long bridge. The shape is described here.