For rule 30 and rule 150 it is exactly 1/2, while for rule 182 it is 3/4. … But in rule 126, black cells always tend to appear in pairs, while in rule 182, every white cell tends to be surrounded by black ones.

The pictures at the top of the page show the base 2 digit sequences of successive numbers obtained by iterating this mapping, while the pictures in the middle of the page plot the sizes of these numbers.

[Construction of] universal objects A more direct way to create a universal object is to set up, say, a 4D array in which two of the dimensions range respectively over possible 1D cellular automaton rules and over possible initial conditions, while the other two dimensions correspond to space and time in the evolution of each cellular automaton from each initial condition.

But an important difference is that while in the many-worlds approach, branchings are associated with possible observation or measurement events, what I suggest here is that they could be an intrinsic feature of even the very lowest-level rules for the universe.

For rule 30, the peak repetition periods are of order 2 0.63 n , while for rule 45, they are close to 2 n (for n = 29 , for example, the period is 463,347,935, which is 86% of the maximum possible).

But after looking at the picture for a while, the boundaries between the patches seem to get somewhat clearer.

A black element in the tag system is set up to correspond to a block of four cells in the Turing machine, while a white element corresponds to a single cell.

For rule 254, for example, it is 8t + 2 , while for rule 90 it is 4t + 2 . … (Some practical systems are found to yield fairly small BDDs, while others are not.)

A practical issue that arises is that if one repeatedly tries experiments with the same set of textures, then after a while one learns to discriminate these particular textures better.

In addition, affine transformations can readily be generalized to any number of dimensions, while complex numbers represent only two dimensions.