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densities of certain local features. And the reason that this happens so quickly when we look at an image is no doubt that the procedure for picking out such features is a very simple one that can readily be carried out in parallel by large numbers of separate cells in our eyes and brains.

For patterns and textures, however, unlike for colors, we can always get beyond the immediate impression that our visual system provides. And so for example, by making a conscious effort, we can scan an image with our eyes, scrutinizing different parts in turn and comparing whatever details we want.

But what kinds of things can we expect to tell in this way? As the pictures below suggest, it is usually quite easy to see if an image is purely repetitive—even in cases where the block that repeats is fairly large.

But with nesting the story is quite different. All eight pictures on the facing page were generated from the two-dimensional substitution systems shown, and thus correspond to purely nested patterns. But except for the last picture on each row—which happen to be dominated by large areas of essentially uniform color—it is remarkably difficult for us to tell that the patterns are nested. And this can be viewed as a clear example of a limitation in our powers of visual perception.

As we found two sections ago, many standard methods of data compression have the same limitation. But at the end of that section I showed that the fairly simple procedure of two-dimensional pointer

Captions on this page:

Examples of all the distinct repetitive patterns that can be formed from arrays of 2×2 and 3×3 blocks. In every single case the presence of pure repetition is easy to recognize by eye. Note that in a pattern generated by repeating one particular block, there will normally be other blocks that occur with other alignments. Page 215 shows patterns obtained in systems based on constraints in which one effectively requires that only certain blocks or sets of blocks occur.

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