Notes

Chapter 5: Two Dimensions and Beyond

Section 4: Substitution Systems and Fractals


Connection [of 2D substitution systems] with digit sequences

Just as in the 1D case discussed on page 891, the color of a cell at position {i, j} in a 2D substitution system can be determined using a finite automaton from the digit sequences of the numbers i and j. At step n, the complete array of cells is

Table[If[FreeQ[Transpose[IntegerDigits[{i, j}, k, n]], form], 1, 0], {i, 0, k^n-1}, {j, 0, k^n-1}]

where for the pattern on page 187, k = 2 and form = {0, 1}. For patterns (a) through (f) on page 188, k = 3 and form is given respectively by (a) {1, 1}, (b) {0|2, 0|2}, (c) {0|2, 0|2} | {1, 1}, (d) {i_, j_} /; j > i, (e) {0, 2} | {1, 1} | {2, 0}, (f) {0, 2} | {1, 1}. Note that the excluded pairs of digits are in exact correspondence with the positions of which squares are 0 in the underlying rules for the substitution systems. (See pages 608 and 1091.)


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