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encoding will succeed in recognizing nesting. So it is not that nesting is somehow fundamentally difficult to recognize; it is just that the particular processes that happen to occur in human visual perception do not in general manage to do it.

So what about randomness? The pictures on the next page show a few examples of images with various degrees of randomness. And just by looking at these images it is remarkably difficult to tell which of them is in fact the most random.

The basic problem is that our visual system makes us notice local features—such as clumps of black squares—even if their density is consistent with what it should be in a completely random array. And as a result, much as with constellations of stars, we tend to identify what seem to be regularities even in completely random patterns.

In principle it could be that there would be images in which our visual system would notice essentially no local features. And indeed in

Captions on this page:

Examples of nested patterns created by following the two-dimensional substitution rules shown. Except for the last examples on each row, it is remarkably difficult to recognize the nested structure in these patterns by eye, even with quite careful scrutiny. The two-dimensional pointer-based encoding scheme from page 571 does however manage to recognize the structure in all cases.

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