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In different situations the reasons for using such probabilistic models have been somewhat different, but before the discoveries in this book one of the key points was that it seemed inconceivable that there could be deterministic models that would reproduce the kinds of complexity and apparent randomness that were so often seen in practice.
And in the first case on the facing page , it so happens that the sequence of digits for each of the initial points shown is indeed quite random, so the behavior we see is correspondingly random.
But one of the crucial points discovered in this book is that more complex phenomena do not always require more complex models.
With short initial conditions, the pictures at the top of the next page demonstrate that combinators tend to evolve quickly to simple fixed points.
In Chladni figures of sand on vibrating plates and in cloud streets in the atmosphere what happens is that material collects at points of zero displacement.
But the presence of the Mod takes the points off this line whenever a n[i] ≥ m . … And that is why in the case a = 3 the points in the plot fall on obvious lines.
In the case a = 65539 , the points lie on planes in 3D.
Thus, for example, a cellular automaton can readily be set up to represent the effect of an inhibition on growth at points on the surface of a snowflake where new material has recently been added.
And the connections between nodes in the network can then be thought of as defining the pattern of neighbors for points in spacetime.
And so when clusters of nodes that are nearby with respect to connections on the network get updated, they can potentially propagate effects to what might be considered distant points in space.
But one of the starting points for the science in this book is that when it comes to more complex behavior mathematics has never in fact done well at explaining most of what we see every day in nature.