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.

Introduction

When confronted with a number sequence, the first thing is to try to conjecture a rule or formula, and then (the hard part) prove that the formula is correct. This chapter had its origin in the study of one such sequence, 1, 8, 8, 24, 8, 64, 24, 112, 8, 64, 64, 192, … (A1602391), although several similar sequences will also be discussed.

These sequences arise from studying how activity spreads in cellular automata (for background see [2, 5, 8, 11, 14, 17, 20, 21, 23, 24, 26]). If we start with a single ON cell, how many cells will be ON after n generations? The preceding sequence arises from the CA known as Fredkin's Replicator [13]. In 2014, Hrothgar sent the author a manuscript [10] studying this CA, and conjectured that the sequence satisfied a certain recurrence. One of the goals of the present chapter is to prove that this conjecture is correct; see (2.31).

In §2.2 we discuss a general class (the “odd-rule” CAs) to which Fredkin's Replicator belongs, and in §2.3 we introduce an operation on number sequences (the “run length transform”) that helps in understanding the resulting sequences. Fredkin's Replicator, which is based on the Moore neighborhood, is the subject of §2.4, and §2.5 analyzes another odd-rule CA, based on the von Neumann neighborhood with a center cell. Although these two CAs are similar, different techniques are required for establishing the recurrences. Both proofs involve making scissor cuts to dissect the configuration of ON cells into recognizable pieces.