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With a pattern that is purely repetitive, the formula is always straightforward, as the picture at the bottom of the facing page illustrates. For all one ever need do is to work out the remainder from dividing the position of a particular square by the size of the basic repeating block, and this then immediately tells one how to look up the color one wants.

So what about nested patterns? It turns out that in most of traditional mathematics such patterns are already viewed as quite advanced. But with the right approach, it is in the end still fairly straightforward to find formulas for them.

The crucial idea—much as in Chapter 4—is to think about numbers not in terms of their size but instead in terms of their digit sequences. And with this idea the picture on the next page shows an example of how what is in effect a formula can be constructed for a nested pattern.

What one does is to look at the digit sequences for the numbers that give the vertical and horizontal positions of a certain square. And then in the specific case shown one compares corresponding digits in these two sequences, and if these digits are ever respectively 0 and 1, then the square is white; otherwise it is black.

So why does this procedure work?

As we have discussed several times in this book, any nested pattern must—almost by definition—be able to be reproduced by a neighbor-independent substitution system. And in the case shown on the next page the rules for this system are such that they replace each square at each step by a 2×2 block of new squares. So as the picture illustrates this means that new squares always have positions that involve numbers containing one extra digit. With the particular rules shown, the new squares always have the same color as the old one, except in one specific case: when a black square is replaced, the new square that appears in the upper right is always white. But this square

Captions on this page:

An example of how the color of any square in a repetitive pattern can be found from its coordinates by a simple mathematical procedure. The procedure takes the x and y coordinates of the square, and computes their remainders after division by 3 and 2 respectively. Using these remainders—which are shown inside each square—the color of a particular square can be determined by a simple lookup in the repeating block shown on the left. The whole procedure can be represented using a mathematical formula that involves either functions like Mod or more traditional functions like Sin.

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