Complex dynamics emerging in Rule 30 with majority memory

Abstract. We use the grid consisting of bits of 3n to motivate the definition of 2-adic numbers. Specifically, we exhibit diagonal stripes in the bits of 32n, which turn out to be the first in an infinite sequence of such structures. Our observations are explained by a 2-adic power series, providing some regularity among the disorder in the bits of powers of 3. Generally, the base-p representation of kpn has these features. 1. Several mysteries The binary representation of a number m can be thought of as encoding the unique set of distinct powers of 2 that sum to m. For example,