# Number-conserving cellular automata with a von Neumann neighborhood of range one

## TL;DR

This paper establishes necessary and sufficient conditions for number-conserving cellular automata with von Neumann neighborhoods of range one across any dimension, enabling efficient analysis of such automata's conservation properties.

## Contribution

It provides a comprehensive geometric framework for characterizing number-conserving cellular automata in any dimension and state set, extending previous results.

## Key findings

- Derived necessary and sufficient conditions for number conservation
- Applicable to any dimension and state set containing zero
- Conditions are computationally tractable in higher dimensions

## Abstract

We present necessary and sufficient conditions for a cellular automaton with a von Neumann neighborhood of range one to be number-conserving. The conditions are formulated for any dimension and for any set of states containing zero. The use of the geometric structure of the von Neumann neighborhood allows for computationally tractable conditions even in higher dimensions.