Notes

Chapter 5: Two Dimensions and Beyond

Section 4: Substitution Systems and Fractals


Connection [of geometric substitution systems] with digit sequences

Patterns after t steps can be viewed as containing all t-digit integers in an appropriate complex base. Thus the patterns on page 189 can be formed from t-digit integers in base I-1 containing only digits 0 and 1, as given by

Table[FromDigits[IntegerDigits[s, 2, t], I - 1], {s, 0, 2^t-1}]

In the particular case of base I-q with digits 0 through q^2, it turns out that for sufficiently large t any complex integer can be represented, and will therefore be part of the pattern. (Compare page 1094.)

From Stephen Wolfram: A New Kind of Science [citation]