More particles [in physics] To produce more massive particles requires higher-energy particle collisions, and today's accelerators only allow one to search up to masses of perhaps 200 GeV. … I am not sure whether in my approach one should expect an infinite series of progressively more massive particles. … But with more complicated models one can avoid this constraint.

(Also considered is strong confluence—that paths can always converge in at most one step, and local confluence—that paths can converge after diverging for one step but not necessarily more. … One can say that two strings are equivalent if they can both be transformed to the same string by using the rules of the multiway system. … If f is Flat , then in evaluating f[a, b, c] one can equally well start with f[f[a, b], c] or f[a, f[b, c]] .

Self-similarity [of rule 90] The pattern generated by rule 90 after a given number of steps has the property that it is identical to what one would get by going twice as many steps, and then keeping only every other row and column. … In the limit of an infinite number of steps one gets a fractal known as a Sierpiński pattern (see page 934 ), with fractal dimension Log[2, 3] ≃ 1.59 (see page 933 ).

But one can also set up generalizations in which the cells correspond to nodes in arbitrary networks. Given a network of the kind discussed in the main text of this section, one can assign a color to each node, and then update this color at each step according to a rule that depends on the colors of the nodes to which the connections from that node go.

One exception is rule 73, which yields densities that continue to oscillate with a period of 3 steps forever. … The boundaries between regions come from blocks of even numbers of black cells in the initial conditions, and if one does not allow any such blocks, the density oscillations no longer occur.

As a rather different approach, one can consider a fixed underlying rule—say a class 4 cellular automaton—with modifications in initial conditions. … As an alternative to modelling individual organisms, one can also consider substitution systems which directly generate genealogical trees for populations of organisms, somewhat like Leonardo Fibonacci 's original model of a rabbit population.

But given presentations of two knots, no finite procedure is known that determines in general whether the knots are equivalent (or constructs a sequence of Reidemeister moves that transform one into the other). Quite probably this is in general undecidable, though since the 1920s a few polynomial invariants have been discovered—with recent ones being related to ideas from quantum field theory—that have allowed some progress to be made.

Rule 30 has much more complicated behavior and yields Boolean expressions whose size grows rapidly from one step to the next.

Comparison to multiway systems Operator systems are normally based on equations, while multiway systems are based on one-way transformations. … With slightly more effort multiway systems with ordinary one-way rules can also be converted to operator systems. … As discussed on page 898 , one can set up operator evolution systems similar to symbolic systems (see page 103 ) that have essentially the same relationship to operator systems as sequential substitution systems do to multiway systems.

And for this to happen s or t must match some part w of u or v . The simplest way this can be achieved is for s or t to reproduce w when its variables are replaced by appropriate expressions. But in general one can make replacements not only for variables in s and t , but also for ones in w .