On the Number of ON Cells in Cellular Automata

TL;DR

This paper investigates the number of ON cells in cellular automata starting from a single ON cell, introducing new sequence transformations and analysis techniques for certain rules, with implications for understanding CA growth patterns.

Contribution

It introduces the run length transform and analytical methods to determine ON cell counts in specific cellular automata rules, advancing the understanding of their growth behavior.

Findings

Exact formulas for ON cell counts in Rule 150, Rule 614, and Fredkin's Replicator

Use of run length transform to analyze CA sequences

Analysis of pattern complexity in various cellular automata

Abstract

If a cellular automaton (CA) is started with a single ON cell, how many cells will be ON after n generations? For certain "odd-rule" CAs, including Rule 150, Rule 614, and Fredkin's Replicator, the answer can be found by using the combination of a new transformation of sequences, the run length transform, and some delicate scissor cuts. Several other CAs are also discussed, although the analysis becomes more difficult as the patterns become more intricate.