But while this process does quite often yield programs with complex behavior, I argued earlier in this book that it does not usually manage to mold anything but fairly simple aspects of this behavior. … But while this is enough to see a tremendous range of behavior, there is no guarantee that one will in fact run across whatever specific features one is looking for.

Other integer functions IntegerExponent[n, k] gives nested behavior as for decimation systems on page 909 , while MultiplicativeOrder[k, n] and EulerPhi[n] yield more complicated behavior, as shown on pages 257 and 1093 .

The first curve yields a perfect fifth, while the third curve yields a diminished fifth (or tritone) in an equal temperament scale.

If this total is less than 4, then the cell becomes white on the next step, while if the total is greater than 6, it becomes black.

(Note that in a 5×5 array, there are 10 interior squares that are subject to the constraints, while in a 10×10 array there are 65.)

And what we see immediately from these pictures is that while some systems exhibit exactly the kind of randomization implied by the Second Law, others do not. … And for a while what one sees is that the randomness that has been inserted persists.

And unlike for example electric charge the observed masses of elementary particles never seem to be in simple ratios—so that for example the muon is about 206.7683 times the mass of the electron, while the tau lepton is about 16.819 times the mass of the muon. … A defining feature of any particle is that it can somehow move in space while maintaining its identity.

One issue—beyond the obvious fact that sounds cannot be included directly in a printed book—is that while one can study the details of a picture at whatever pace one wants, a sound is in a sense gone as soon as it has finished playing. … Musical notes usually have just one basic frequency, while voiced speech sounds have two or three.

But while picture (c) shows the effects of various lines carrying information around the system, it gives no indication of why the lines should behave in the way they do. … The beginning of each block is indicated by a dashed line, while the elements within the block are indicated by solid black and gray lines.

The reason for this difference is that in rule 22 the randomness we see is intrinsically generated by the evolution of the system, while in rule 90 it comes from randomness in the initial conditions.