[Cognitive] child development As children get older their thinking becomes progressively more sophisticated, advancing through a series of fairly definite stages that appear to be associated with an increasing ability to handle generalization and abstraction.

For rule 30, the numbers of steps needed for each block of lengths 1 through 10 to appear at least once is {1, 2, 4, 12, 22, 24, 33, 59, 69, 113} .

As discussed on page 903 , this 1D sequence can be obtained by looking at how a line with GoldenRatio slope cuts through a 2D lattice of squares. Penrose tilings can be obtained by looking at how a 2D plane with slopes based on GoldenRatio cuts through a lattice of hypercubes in 5D.

But the experiments I have carried out do suggest that, just as with simple register machines, searching through many millions of short programs typically yields at least a few that exhibit complex and seemingly random behavior.

Typically the network topology of a foam continually rearranges itself through cascades of seemingly random T1 processes (rule (b) from page 511 ), with regions that reach zero size disappearing through T2 processes (reversed rule (a)).

Signals can sometimes travel through the magnetosphere along magnetic field lines from one hemisphere to the other, yielding so-called whistlers with frequencies that fall off in a highly regular way with time. … Variations of source properties on timescales of months or years are not uncommon; variations of signals on timescales of tens of minutes can be introduced by propagation through turbulence in the interstellar medium. … When they first form after a supernova pulsars have millisecond repetition rates, but over the course of a few million years they slow to repetition rates of seconds through a series of glitches, associated perhaps with cracking in their solid crusts or perhaps with motion of quantized vortices in their superfluid interiors.

In each case a slice is taken through the midline of the box.

Thus the patterns on page 189 can be formed from t -digit integers in base  - 1 containing only digits 0 and 1, as given by Table[FromDigits[IntegerDigits[s, 2, t],  - 1], {s, 0, 2 t -1}] In the particular case of base  - q with digits 0 through q 2 , it turns out that for sufficiently large t any complex integer can be represented, and will therefore be part of the pattern.

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.

There are a total of 7,625,597,484,987 cellular automata with three colors and nearest-neighbor rules, and searching through these one finds just 1800 that are reversible.