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Pandas and anteaters, for example, typically have just a few regions of different color, while other animals can have hundreds of regions.
Of the 274,499 axiom systems of the form {… a} where … involves ∘ up to 6 times, 32,004 allow only operators {6,9} , while 964 allow only {1,7} .
Typically they imagine that while the pictures may look complicated, they would actually seem simple if only they were subjected to the appropriate kind of analysis.
In 4D the six regular polytopes have 4, 4, 6, 8, 4 and 12 edges meeting at each vertex, and in higher dimensions the simplex ( d + 1 vertices) and hypercube ( 2 d vertices) have d edges meeting at each vertex, while the co-cube ( 2d vertices) has 2(d - 1) .
But while we have discussed a whole range of different kinds of underlying rules, we have for the most part considered only the simplest possible initial conditions—so that for example we have usually started with just a single black cell.
In case (b), it can move between 0, 1 or 2 positions, while in case (c) it can move any distance between 0 and 1 at each step.
In both examples, what essentially happens is that in regions where there is an excess of black over white, an increasingly large fraction of cells become black, while in regions where there is an excess of white over black, the reverse happens.
The pictures below show three examples of what can happen, while the facing page shows the effects of systematically varying certain growth rates.
So is it possible to get more complex behavior while maintaining reversibility?
In the first kind of object, the actual sequence is then given, while in the second kind of object what is given is a specification of how far back in the data the required sequence can be found.