The pictures below show the behavior of several sequential cellular automata with k = 2 , r = 1 elementary rules. … The following will update triples of cells in the specified order by using the function f : OrderedUpdate[f_, a_, order_]:= Fold[ReplacePart[ #1, f[Take[#1, {#2 - 1, #2 + 1}]], #2] &, a, order] A random ordering of n cells corresponds to a random permutation of the form Fold[Insert[#1, #2, Random[Integer, Length[#1]] + 1] &, {}, Range[n]]

This is a k = 8 2D cellular automaton in which toppling of sand above a critical slope is captured by updating an array of relative sand heights s according to the rule SandStep[s_]:= s + ListConvolve[ {{0, 1, 0}, {1, -4, 1}, {0, 1, 0}}, UnitStep[s - 4], 2, 0] Starting from any initial condition, the rule eventually yields a fixed configuration with all values less than 4, as in the picture below. … In 1D, the update rule is simply SandStep[s_] := s + ListConvolve[{1, -2, 1}, UnitStep[s - 2], 2, 0] In this case the evolution obtained if one repeatedly adds to the center cell (as in the first picture below) is always quite simple. … When d > 1 , more complicated behavior is seen for evolution from at least some initial conditions, as indicated above.

Turing machines [from cellular automata] Given any Turing machine with rules in the form used on page 888 and k possible colors for each cell, a cellular automaton which emulates it can be constructed using TMToCA[rules_, k_:2] := Flatten[{Map[g[#, k]&, rules], {_, x_, _}  x}] g[{s_, a_}  {sp_, ap_, d_}, k_] := {If[d  1, Identity, Reverse][{k s + a, x_, _}]  k sp + x, {_, k s + a, _}  ap} If the Turing machine has s states for its head, then the cellular automaton has k (s+1) colors for each cell. An initial condition with a single cell of color k surrounded by 0's corresponds to being in state 1 with a blank tape in the Turing machine.

An example suggested by Stanislaw Ulam around 1960 (in a peculiar attempt to get a 1D analog of a 2D cellular automaton; see pages 877 and 928 ) starts with {1, 2} , then successively appends the smallest number that is the sum of two previous numbers in just one way, yielding {1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, 48, 53, 57, …} With this initial condition, the sequence is known to go on forever.

The combination Drop[list, -1] + 2 Drop[list, 1] of the result from CA2EvolveList corresponds to evolution according to a first-order k = 4 , r = 1 rule.

If one value is n , then the next value is 3n/2 if n is even, and (3n+1)/2 if n is odd. The initial condition is n=1 .

The zeta function Zeta[s] is defined as Sum[1/k s , {k, ∞ }] . The curve shown here is the so-called Riemann–Siegel Z function, which is essentially Zeta[1/2 +  t] .

In case (b), it can move between 0, 1 or 2 positions, while in case (c) it can move any distance between 0 and 1 at each step.

has the property that its vertical position ends with a 0, and its horizontal position ends with a 1. … If any digit in the y coordinate of a particular square is 0 when the corresponding digit in the x coordinate is 1 then the square is white; otherwise it is black.

Simple case [of three-body problem] The position of the idealized planet in the case shown satisfies the differential equation δ tt z[t]  -z[t]/(z[t] 2 + (1/2 (1 + e Sin[2 π t] ) 2 ) 3/2 where e is the eccentricity of the elliptical orbit of the stars ( e = 0.1 in the picture). … Following work by Kirill Sitnikov in 1960 and by Vladimir Alekseev in 1968, it was established that with suitably chosen initial conditions, the equation yields any sequence Floor[t[i + 1] - t[i]] of successive zero-crossing times t[i] .