On the communication between cells of a cellular automaton on the penta- and heptagrids of the hyperbolic plane

TL;DR

This paper explores communication methods in cellular automata on hyperbolic penta- and heptagrids, introducing a coordinate system and message exchange improvements for efficient simulation of hyperbolic cellular automata.

Contribution

It presents a new coordinate system and message exchange method for cellular automata on hyperbolic grids, enhancing simulation efficiency and communication speed.

Findings

Coordinates enable precise tile location in hyperbolic tilings

Communication along geodesics improves message exchange speed

Linear relation between automaton state and cell coordinates

Abstract

This contribution belongs to a combinatorial approach to hyperbolic geometry and it is aimed at possible applications to computer simulations. It is based on the splitting method which was introduced by the author and which is reminded in the second section of the paper. Then we sketchily remind the application to the classical case of the pentagrid, i.e. the tiling of the hyperbolic plane which is generated by reflections of the regular rectangular pentagon in its sides and, recursively, of its images in their sides. From this application, we derived a system of coordinates to locate the tiles, allowing an implementation of cellular automata. At the software level, cells exchange messages thanks to a new representation which improves the speed of contacts between cells. In the new setting, communications are exchanged along actual geodesics and the contribution of the cellular automaton is also linear in the coordinates of the cells.