The author explores the concept of binary tiling of the hyperbolic plane, discussing various aspects such as shrinking vertical edges, a unique tessellation, connections to Smith charts and Escher, 3-coloring tiles, a half-flipped variation, and applications in origami. An intriguing method of numbering tiles is shared, demonstrating how the position of tiles can be encoded with binary sequences. The discussion delves into the concept of uncountable binary tilings, revealing a link to the exotic number system of 2-adic integers. Various properties of the tiling are tied to properties of these numbers, providing a deep insight into the intricate nature of binary tilings.
https://11011110.github.io/blog/2024/10/28/2-adic-numbering-binary.html