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In the upper block of pictures, every cell is chosen to be black or white with equal probability on the two successive first steps.

Sometimes the actual repetition period is equal to this maximum value. … The bottom plot gives the repetition period for this system as a function of its size; for odd n this period is equal to MultiplicativeOrder[2, n] .

As we will see on page 338 the presence of such patterns is particularly clear when there are equal numbers of black and white cells in the initial conditions—but how these cells are arranged does not usually matter much at all. … The presence of such structure is most obvious when there are equal numbers of black and white cells in the initial conditions, but it does not rely on any regularity in the arrangement of these cells.

As in rule 184, the nested behavior seen here is most obvious when the density of black and white cells in the initial conditions is equal.

An example of the three-body problem, in which an idealized planet moves up and down through the plane of two equal-mass idealized stars in a perfect elliptical orbit. … The planet is assumed to be of negligible mass relative to the stars, and to start with zero vertical velocity at exactly an equal distance between the stars.

In physics there also seems to be a maximum speed at which the effects of any event can spread: the speed of light, equal to about 300
A very simple substitution system whose causal network has slices that can be thought of as corresponding to a highly regular idealization of one-dimensional ordinary space. … Examples of patterns produced by cellular automata, illustrating the fact discussed in Chapter 6 that the edge of each pattern has a maximum slope equal to one cell per step, corresponding to an absolute upper limit on the rate of information transmission—similar to the speed of light in physics.

Nesting in rule 184 is easiest to see when the initial conditions contain exactly equal numbers of black and white cells, so that the numbers of left and right stripes exactly balance, and all stripes eventually annihilate. … The initial condition used has exactly equal numbers of black and white cells, causing all the stripes eventually to annihilate.

And in general it is fairly easy to see that in any sequence that is purely repetitive there must beyond a certain length be many blocks whose frequencies are far from equal.
… (Note that these substitution systems are the simplest ones that yield equal frequencies of all blocks up to lengths 1, 2 and 3 respectively.)

If all k letters have equal probabilities, there will be many words with equal frequency, so the distribution will contain steps, as in the second picture below. … If all letter probabilities are equal, then words will simply be ranked by length, with all k m words of length m occurring with frequency p m .

Particles are inserted in a regular way at the left-hand end so as to maintain an overall flow speed equal to about 0.4 of the maximum possible.