Mathematical Tools for Patterns I've also looked into the mathematical tools used to study pattern formation. Reaction-diffusion equations can explain how patterns arise from the interaction and movement of substances. Cellular automata show how simple rules at a local level can lead to complex global patterns. Network theory helps analyze the structure and behavior of interconnected systems, and fluid dynamics describes how patterns emerge in liquids and gases. Planetary Ring Insights I've gathered more information about planetary rings. They appear to be made of countless small particles, mostly ice, orbiting a planet. The formation could be from a moon breaking apart due to collisions or tidal forces, or from material that never coalesced into a moon. Their structure is influenced by collisions between particles, the gravitational effects of small 'shepherd' moons, and resonances with larger moons.