Webtagr - Technology News Summarizer

Collatz's Ant and Landscape Similarity: The Mystery of Beta

2025-05-28

This post explores the similarity of landscapes generated by Collatz's ant trajectories. By analyzing stopping time (τ), maximum Euclidean distance (α), the step at which the maximum distance is reached (β), and the final distance (γ), the author finds that stopping time is not a decisive factor in landscape similarity. While maximum distance (α) is related to landscape scale, it's insufficient to distinguish different landscapes. However, the step at which the maximum distance is reached (β) appears to be an indicator for distinguishing different landscapes, but the underlying mechanism requires further investigation. The article presents multiple examples showing the complex relationship between β and landscape shape and poses some unsolved mysteries, such as why, when the maximum distance (α) is different, β is sometimes the same and sometimes different? This provides a new perspective on the study of the Collatz conjecture.

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

2025-01-13

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.

Collatz's Ant: Visualizing Collatz Sequences with Langton's Ant

2024-12-23

Collatz's Ant visualizes Collatz sequences using Langton's Ant rules. Based on the Collatz function (even numbers halved, odd numbers multiplied by 3 and added to 1), the ant turns 90 degrees clockwise for even numbers and counter-clockwise for odd numbers. The cell's state flips with each move, repeating until n=1. Code and examples demonstrate consecutive trajectories from 10^30 to 10^30+20.