It appears that only when the group is nilpotent (so that certain combinations of elements commute much as they do on a lattice) is there polynomial growth in the Cayley graph and thus finite dimension.

Alternative rules [for cryptography] Among elementary rules, rule 45 is the only plausible alternative to rule 30. … Among the 4,294,967,296 r = 2 rules which depend on 5 cells, there are again just a few that give long periods, but now only a small fraction of these seem directly related to rules 45 and 30, and perhaps half are not additive with respect to any position.

In the simplest case, studied especially in the context of perceptrons in the 1960s, one has only two sets of neurons: an input layer and an output layer. … But out of the 2 2 n possible Boolean functions of n inputs, only 14 (out of 16) can be obtained for n = 2 , 104 (out of 256) for n = 3 , 1882 for n = 4 , and 94304 for n = 5 . … But in special circumstances probably not of great biological relevance it can yield class 2 behavior.

And particularly with the rise of quantum mechanics it came to be thought that meaningful calculations could be done only on probabilities, not on individual random sequences. … For not knowing about the phenomenon of intrinsic randomness generation, it has normally been assumed that with the kinds of discrete elements and fairly simple rules common in such models, realistically complicated behavior can only ever be obtained if explicit randomness is continually introduced.

Already in 1882 George FitzGerald and Hendrik Lorentz noted that if there was a contraction in length by a factor Sqrt[1 - v 2 /c 2 ] in any object moving at speed v (with c being the speed of light) then this would explain the result. … Then in 1905 Albert Einstein proposed his so-called special theory of relativity—which took as its basic postulates not only that the laws of mechanics and electrodynamics are independent of how fast one is moving, but that this is also true of the speed of light. … In the late 1800s Ernst Mach had emphasized the idea of formulating science and particularly mechanics in terms only of concepts that can actually be measured by observers.

But most of the scientific work that was done ended up being based only on my earliest discoveries, and being very much within the framework of one or another of the existing sciences—with the result that it managed to make very little progress on any general and fundamental issues.

One might have thought that perhaps the basic phenomenon of complexity that I have identified could only occur in discrete systems.

And what one sees is that it takes only a small change in the initial speed to make the ball come to rest in a completely different orientation.

For essentially all of them involve only discrete elements which can be handled quite directly on a practical computer.

And it takes only very simple rules to generate shapes that look like the villi and other corrugated structures one often sees in animals. … The formulas for local curvature as a function of arc length for each set of pictures are as follows: 1 (circle); s (Cornu spiral or clothoid); s 2 ; 1/Sqrt[s] (involute of circle); 1/s (logarithmic or equiangular spiral); 1/s 2 ; Exp[-s 2 ] ; Sin[s] ; s Sin[s] .