The details of these trajectories cannot be deduced quite as directly as before from the digit sequences of initial positions, but Paths followed by four idealized balls dropped from initial positions differing by one part in a thousand into an array of identical circular pegs.

The problem turns out to be a rather general one.

But there are certainly cases—in one dimension and particularly in the first set of images below—where different domains do combine, and exact repetition is achieved.

One might have thought that continuum behavior would somehow rely on special features of actual systems in physics.

The basic idea is to have the active cell in the mobile automaton sweep backwards and forwards, updating cells as it goes, in such a way that after each complete sweep it has effectively performed one step of cellular automaton evolution.

(One sees many branches in the fossil record—such as organisms with dominant symmetries other than fivefold—but all seem to have the same ancestry.) … In the 1970s it then became popular to investigate complicated cycles of chemical reactions that seemed analogous to ones found in living systems.

Note that the fact that a number is normal in one base does not imply anything about its normality in another base (unless the bases are related for example by both being powers of 2). … One based on gradual extension of work by Richard Stoneham from 1971 is that numbers of the form Sum[1/(p n b p n ), {n, ∞ }] for prime p > 2 are normal in base b (for GCD[b, p]  1 ), and are transcendental.

One can characterize the symmetry of a pattern by taking the list v of positions of cells it contains, and looking at tensors of successive ranks n : Apply[Plus, Map[Apply[Outer[Times, ##] &, Table[#, {n}]] &, v]] For circular or spherical patterns that are perfectly isotropic in d dimensions these tensors must all be proportional to (d - 2)!!… (In general there will be more than one component unless the representation of the lattice symmetry group carried by the rank n tensor is irreducible.) … Even though it is not inevitable from lattice symmetry, one might think that if there is some kind of effective randomness in the underlying rules then sufficiently large patterns would still often show some sort of average isotropy.

That what one would usually call complexity can be present in mathematical systems was for example already noted in the 1890s by Henri Poincaré in connection with the three-body problem (see page 972 ). … One attempt at an abstract definition was what Charles Bennett called logical depth: the number of computational steps needed to reproduce something from its shortest description. … In general it is possible to imagine setting up all sorts of definitions for quantities that one chooses to call complexity.

One can also consider building up lists of non-identical elements, say by successively using Join .