Simulations between triangular and hexagonal number-conserving cellular automata

TL;DR

This paper explores the conditions under which triangular and hexagonal cellular automata conserve numbers, providing a framework for simulating one type with the other based on flow functions.

Contribution

It introduces necessary conditions for number conservation in triangular and hexagonal cellular automata and demonstrates how to simulate one with the other using flow functions.

Findings

Derived necessary conditions for number conservation.

Expressed transition functions as sums of flow functions.

Constructed effective simulations between automata types.

Abstract

A number-conserving cellular automaton is a cellular automaton whose states are integers and whose transition function keeps the sum of all cells constant throughout its evolution. It can be seen as a kind of modelization of the physical conservation laws of mass or energy. In this paper, we first propose a necessary condition for triangular and hexagonal cellular automata to be number-conserving. The local transition function is expressed by the sum of arity two functions which can be regarded as 'flows' of numbers. The sufficiency is obtained through general results on number-conserving cellular automata. Then, using the previous flow functions, we can construct effective number-conserving simulations between hexagonal cellular automata and triangular cellular automata.