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The trajectories obtained with four possible initial positions for the planet—differing by 10 -8 —are shown. … The divergence of trajectories with slightly different initial vertical positions indicates sensitive dependence on initial conditions.
In case (a), each particle moves just one position to the left or right at each step. 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.
In each case the position of a cell is specified by a pair of numbers given as base 2 digit sequences in the initial conditions for a cellular automaton. The evolution of the cellular automaton then quickly determines what the color of the cell at that position in the pattern on the left will be.
Let the density of black cells at position x and time t be f[x,t] , where this density can conveniently be computed by averaging over many instances of the system. If we assume that the density varies slowly with position and time, then we can make series expansions such as f[x + dx, t]  f[x , t] + ∂ x f[x, t] dx + 1/2 ∂ xx f[x, t] dx 2 + … where the coordinates are scaled so that adjacent cells are at positions x - dx , x , x + dx , etc. If we then assume perfect underlying randomness, the density at a particular position must be given in terms of the densities at neighboring positions on the previous step by f[x, t + dt]  p 1 f[x - dx, t] + p 2 f[x, t] + p 3 f[x + dx, t] Density conservation implies that p 1 + p 2 + p 3  1 , while left-right symmetry implies p 1  p 3 .
The maps shown can be thought of as being made by taking an infinitely dense limit of the array of pictures on the facing page , but keeping only what one sees in each picture by looking through a peephole at a particular position relative to the original stem.
Implementation [of finite automata for nested patterns] Given the rules for a substitution system in the form used on page 931 a finite automaton (as on page 957 ) which yields the color of each cell from the digit sequences of its position is Map[Flatten[MapIndexed[#2 - 1  Position[rules, #1  _] 〚 1, 1 〛 &, Last[#], {-1}]] &, rules] This works in any number of dimensions so long as each replacement yields a block of the same cuboidal form.
Practical empirical mathematics In looking for formulas to describe behavior seen in this book I have in practice typically taken associated sequences of numbers and then tested whether obvious regularities are revealed by combinations of such operations as: computing successive differences (see note below ), computing running totals, looking for repeated blocks, picking out running maxima, picking out numbers with particular modular residues, and looking at positions of particular values, and at the forms of the digit sequences of these positions.
Behavior of a simple aggregation model, in which a single new black cell is added at each step at a randomly chosen position adjacent to the existing cluster of black cells.
Note that on each line in each picture, only the order of elements is ever significant: as the insets show, a particular element may change its position as a result of the addition or subtraction of elements to its left.
The positions of the black cells are given by (and this establishes the connection with the picture on page 117 ) Fold[Flatten[{#1 - #2, #1 + #2}] &, {0}, 2^DigitPositions[t]] DigitPositions[n_] := Flatten[Position[Reverse[IntegerDigits[n, 2]], 1]] - 1 The actual pattern generated by rule 90 corresponds to the coefficients in PolynomialMod[Expand[(1/x + x) t ], 2] (see page 1091 ); the color of a particular cell is thus given by Mod[Binomial[t, (n + t)/2], 2] /; EvenQ[n + t] .
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