Particle masses The measured masses of known elementary particles in units of GeV (roughly equal to the proton mass) are: photon: 0, electron: 0.000510998902; muon: 0.1056583569; τ lepton: 1.77705; W : 80.4; Z : 91.19. … When one sets up a quantum field theory one can typically in effect insert various mass parameters for particles. … Then one introduces self-interactions in this field so as to make its stable state be one that has constant nonzero value throughout the universe.

Each line in this diagram corresponds to one localized structure in rule 110.

How can one set up a simple idealization of the transformations on symbolic expressions that Mathematica does? One approach suggested by the idea of combinators from the 1920s is to consider expressions with forms such as e[e[e][e]][e][e] and then to make transformations on these by repeatedly applying rules such as e[x_][y_]  x[x[y]] , where x_ and y_ stand for any expression.

The procedure is only slightly more complicated than the one for division discussed above. … And it then turns out that the base 2 digits of s correspond exactly to the base 2 digits of √ n —with one new digit being generated at each step.

But overall it seems that the basic kinds of behavior produced are just the same as in one and two dimensions. And in particular, the basic phenomenon of complexity does not seem to depend in any crucial way on the dimensionality of the system one looks at.

In each of these examples the behavior turns out to be rather simple—with for example the number of possible sequences always increasing uniformly from one step to the next. … The plots on the right show the total number of possible states obtained at each step, and the differences of these numbers from one step to the next.

Nevertheless, the pictures on the facing page demonstrate that if one uses initial conditions that are slightly different—though still simple—then one can still see randomness in the behavior of this particular cellular automaton.

And with this setup, if the underlying rules replace each block by one that contains the same number of black cells, it is inevitable that the system as a whole will conserve the total number of black cells. With two possible colors and blocks of size two the only kinds of block cellular automata that conserve the total number of black cells are the ones shown in the second set of pictures—and all of these exhibit rather trivial behavior.

[No text on this page] Examples of a class of one-dimensional constraints where it is in general undecidable whether they can be satisfied. … When the constraints involve more than two blocks there seems in general to be no upper limit on how long a string one may need to consider to tell whether the constraints can be satisfied.

And indeed it turns out that among the 10,552 possible register machines with programs that are four or fewer instructions long, not a single one exhibits more complicated behavior. … It turns out that this program is one of only two (which differ just by interchange of the first and second registers) out of the 248,832 possible programs with five instructions that yield anything other than strictly repetitive behavior.