Visualizing the Collatz Conjecture with Langton's Ant: Similarities and Differences

This post continues a previous one, visualizing the Collatz conjecture using Langton's ant. By simulating the ant's movement on a grid, the author observes that Collatz sequences with similar final patterns often have similar stopping times. However, the inverse isn't true: sequences with the same stopping time can have significantly different trajectories. The author quantifies the similarity between sequences by calculating the size of the intersection of their sets, finding that sequences with small initial number differences have higher similarity, which decreases as the difference grows. This research offers a novel perspective on understanding the complexity of the Collatz conjecture.