The successive steps in the evolution of each substitution system are obtained at the points indicated by arrows.

Symmetric 5-neighbor [2D cellular automaton] rules Among the 32 possible 5-cell neighborhoods shown for example on page 941 there are 12 classes related by symmetries, given by s = {{1}, {2, 3, 9, 17}, {4, 10, 19, 25}, {5}, {6, 7, 13, 21}, {8, 14, 23, 29}, {11, 18}, {12, 20, 26, 27}, {15, 22}, {16, 24, 30, 31}, {28}, {32}} Completely symmetric 5-neighbor rules can be numbered from 0 to 4095, with each digit specifying the new color of the cell for each of these symmetry classes of neighborhoods.

In the first case shown, starting for example at position 4 the dot then visits positions 5, 0, 1, 2 and so on, at each step going from one node in the network to the next. … There are cycles which contain states that are visited repeatedly, and there can also be trees that represent transient states that can each only ever occur at most once in the evolution of the system. … All but one of these 16 states evolve after at most two steps to state 15, which corresponds to all cells being black.

The behavior of a system will be repetitive in time whenever it effectively follows a closed curve—either literally in space, or in terms of states that it visits.

But vastly more common in practice is instability only at specific critical points—say bifurcation points—combined with either intrinsic randomness generation or randomness from the environment.

Specially constructed transcendental numbers Numbers known to be transcendental include ones whose digit sequences contain 1's only at positions n! … Concatenation sequences, as well as generalizations formed by concatenating values of polynomials at successive integer points, are also known to yield numbers that are transcendental.

(The function s[d] has a maximum around d = 5.26 , then decreases rapidly with d .) If instead of flat space one considers a space defined by the surface of a 3D sphere—say with radius a —one can ask about areas of circles in this space. Such circles are no longer flat, but instead are like caps on the sphere—with a circle of radius r containing all points that are geodesic (great circle) distance less than r from its center.

Backtracking [in cellular automata] If one wants to find out which of the 2 n possible initial conditions of width n evolve to yield a specific column of colors in a system like an elementary cellular automaton one can usually do somewhat better than just testing all possibilities. … If one wants to find just a single initial condition that works then one can set up a recursive algorithm that in effect does a depth-first traversal of the tree. No doubt in many cases the number of nodes that have to be visited eventually increases like 2 t , but many branches usually die off quickly, greatly reducing the typical effort required in practice.

Note that if the period is equal to its absolute maximum of m , then every possible n is always visited, whatever n one starts from. … If one could ignore the Mod , then the coordinates would simply be {n[i], a n[i]} , so the points would lie on a single straight line with slope a . … In the case a = 65539 , the points lie on planes in 3D.

In many cases this maximal rectangle overlaps those found at subsequent points.