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But while complexity is visible at this level, it is usually necessary to go to a more detailed level in order to get any real idea of why it occurs.
For even programs with some of the very simplest possible rules yield highly complex behavior, while programs with fairly complicated rules often yield only rather simple behavior.
It takes fewer steps for networks to be built up, but the results are qualitatively similar to those on the previous page : rule (a) yields a nested structure, rule (b) gives repetitive behavior, while rule (c) produces behavior that seems complicated and in some respects random.
And while issues of typical behavior have not really been studied before, it has been known that combinators are universal almost since the concept of universality was first introduced in the 1930s.
And what we will see in this section is while some of these phenomena correspond to known features of mathematics—such as Gödel's Theorem—many have never successfully been recognized.
While humans and primates normally have three types of cone cells, it has been found that other mammals normally have two, while birds, reptiles and fishes typically have between 3 and 5.
Note that while most current works of science tend to refer mainly just to very recent material, this book often refers to material that is centuries or even millennia old—in some ways more in the tradition of fields like philosophy.
But while this may sometimes be true—perhaps as a consequence of the Central Limit Theorem—it is rarely checked, making it likely that many detailed inferences are wrong.
Note that in practice a coin tossed in the air will typically turn over between ten and twenty times while a die rolled on a table will turn over a few tens of times.
Diffusion-limited aggregation (DLA)
While many 2D cellular automata produce intricate nested shapes, the aggregation models shown here seem to tend to simple limiting shapes.