Simulations between triangular and hexagonal number-conserving cellular automata

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.

💡 Research Summary

The paper investigates number‑conserving cellular automata (NCCA) on two non‑rectangular lattices—triangular and hexagonal—and establishes a rigorous framework for proving number‑conservation as well as for constructing efficient simulations between these two lattice types. A number‑conserving cellular automaton is defined as a discrete dynamical system whose cell states are integers and whose global sum of states remains invariant under the local transition rule. While most prior work focused on one‑dimensional or square‑grid two‑dimensional automata, the authors address the challenge posed by lattices where each cell has a different number of neighbours (three for triangular, six for hexagonal).

The core technical contribution is the introduction of a “flow‑function” representation of the local rule. For a cell with neighbourhood ({x_0, x_1, …, x_k}) the new state is expressed as

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