In Chapter 5 we defined the notion of automatic sequence, and in later chapters we explored the properties of these sequences. By definition, an automatic sequences is a one-sided, one-dimensional sequence. But one-dimensional infinite arrays of items are not the only such objects studied in mathematics; two-dimensional arrays (also called tables or double sequences; we use these terms interchangeably) are studied, as well as higher-dimensional objects. In this chapter we will examine a generalization of automatic sequences to a multidimensional setting, concentrating on the two-dimensional case. The interested reader will have no problem extending the results to the multidimensional case.

The Sierpiński Carpet

We start with an example.

Example 14.1.1 Consider the two-dimensional Sierpiński carpet array s = (Si, j)i, j≥0 over {0, 1}, defined as follows: Si, j = 0 if and only if the base-3 expansions of i and j share at least one 1 in an identical position. More precisely, let 0 ≤ i, j < 3n, and let x = an-1 … a0, y = bn-1 … b0 be strings of length n such that [x]3 = i, [y]3 = j. Then Si, j = 0 if and only if there exists an index S, 0 ≤ S < n, such that as = bs = 1.