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But for k > 2 , no such simple characterization appears to exist.
And when there are, for example, two symmetrical sides to a particular system, it is often possible to compare the visual patterns produced on each side, and see what similarities exist.
If at every stage the tissue in each region produced grows at the same rate, and all that differs is what final type of cells will exist in each region, then inevitably a simple and highly regular overall structure will emerge, as in the idealized picture below.
And indeed I will argue in Chapter 12 that the Principle of Computational Equivalence suggests that in general no such reduced procedure should exist.
But remarkably enough there do exist rules for which exactly the same causal network is obtained regardless of what scheme is used.
But despite these restrictions a fairly large number of replacements are still possible; for example, there are a total of 419 distinct ones that exist involving clusters with no more than five nodes.
And at the lowest level what I expect is that even though the rules being applied are perfectly definite, the overall pattern of connections that will exist in the network corresponding to our universe will continually be rearranged in ways complicated enough to seem effectively random.
procedure can indeed be used to check that no purely repetitive pattern exists, but as we will see later in this chapter , it does not successfully detect the presence of even certain highly regular nested patterns.
For a few other nested patterns there exist fairly simple connections with additive cellular automata and similar systems—though usually in more dimensions or with more neighbors.
The basic idea is to have a sequence of layers of nerve cells—much as one knows exist in the brain—with each cell in each successive layer responding only if the inputs it gets from some fixed random set of cells in the layer above form some definite pattern.