Why it is sufficient to consider only the case where the seed of linear cellular automata is $1$

Abstract

When using a cellular automaton (CA) as a fractal generator, consider orbits from the single site seed, an initial configuration that gives only a single cell a positive value. In the case of a two-state CA, since the possible states of each cell are $0$ or $1$, the "seed" in the single site seed is uniquely determined to be the state $1$. However, for a CA with three or more states, there are multiple candidates for the seed. For example, for a $3$-state CA, the possible states of each cell are $0$, $1$, and $2$, so the candidates for the seed are $1$ and $2$. For a $4$-state CA, the possible states of each cell are $0$, $1$, $2$, and $3$, so the candidates for the seed are $1$, $2$, and $3$. Thus, as the number of possible states of a CA increases, the number of seed candidates also increases. In this paper, we prove that for linear CAs it is sufficient to consider only the orbit from the single site seed with the seed $1$.