(For the history of fractals and 2D substitution systems see page 934 .) … The first six levels of the specific pattern in example (a) on page 83 correspond exactly to the segregation diagram for the I Ching that arose in China as early as 2000 BC. … As early as 1851, for example, Eugène Prouhet showed that if sequences of integers were partitioned according to sequence (b) on page 83 , then sums of powers of these integers would be equal: thus Apply[Plus, Flatten[Position[s, i]] k ] is equal for i = 0 and i = 1 if s is a sequence of the form (b) on page 83 with length 2 m , m > k .

The maximum is Nest[2 # &, 0, n] (compare page 906 ), achieved for initial conditions of the form Nest[#[ ℯ ]&, ℯ , n] . … During the evolution the rule can apply only to the inner part FixedPoint[Replace[#, ℯ [x_]  x] &, expr] of an expression. … For initial conditions of size n , this occurs after at most Sum[Nest[2 # &, 0, i] - 1, {i, n}] + 1 steps.

Nested digit sequences The number obtained from the substitution system {1  {1, 0}, 0  {0, 1}} is approximately 0.587545966 in base 10. … From the result on page 890 , the number whose digits are obtained from {1  {1, 0}, 0  {1}} is given by Sum[2^(-Floor[n GoldenRatio]), {n, ∞ }] . … The n th term in its continued fraction representation turns out to be 2^Fibonacci[n - 2] .

Non-periodic pattern [forced by 2D constraint] The color at position x, y in the pattern is given by a[x_, y_] := Mod[y + 1, 2] /; x + y > 0 a[x_, y_] := 0 /; Mod[x + y, 2]  1 a[x_, y_] := Mod[Floor[(x - y) 2 (x + y - 6)/4 ], 2] /; Mod[x + y, 4]  2 a[x_, y_] := 1 - Sign[Mod[x - y + 2, 2 (-x - y + 8)/4 ]] The origin of the x, y coordinates is the only freedom in this pattern. The nested structure is like the progression of base 2 digit sequences shown on page 117 .

But with k = 3 possible elements, there are infinite nested sequences that can, such as the one produced by the substitution system {0  {0, 1, 2}, 1  {0, 2}, 2  {1}} , starting with {0} . … (The number of sequences that work seems to grow roughly like 2 n/2 .) … And for example it is known that for k ≥ 2 any pattern with length 6 or more (excluding the ___ 's) and only two different variables (say x__ and y__ ) can always be avoided.

The magnetic energy of the system is taken to be e[s_] := -1/2 Apply[Plus, s ListConvolve[ {{0, 1, 0}, {1, 0, 1}, {0, 1, 0}}, s, 2], {0, 1}] so that each pair of adjacent spins contributes -1 when they are parallel and +1 when they are not. … (This seems to be true only in 2D, and not in 3D or higher.) … Of the 2 32 general 5-neighbor rules 34 conserve e[s] —but all have only very simple behavior.

On displays consisting of fixed arrays of pixels, gray levels must be obtained by having only a certain density of pixels be black. One way to achieve this is to break the array into 2 n × 2 n blocks, then successively to fill in pixels in each block until the appropriate gray level is reached, as in the pictures below, in an order given for example by Nest[ Flatten2D[{{4 # + 0, 4 # + 2}, {4 # + 3, 4 # + 1}}] &, {{0}}, n] An alternative to this so-called ordered dither approach is the Floyd–Steinberg or error-diffusion method invented in 1976. … But even with m = {1, 0, 1, 0}/2 the method generates fairly random patterns, as in the second row below.

Note that the configuration of relevant cells can be repetitive only if the initial conditions were repetitive (see page 871 ). … Most of the 2 n possible states have unique predecessors; for large n , about 2 0.76 n or Root[# 3 - # 2 - 2 &, 1] n instead have 0 or 2 predecessors. The predecessors of a given state can be found from Cases[Map[Fold[Prepend[#1, If[#2  1 ⊻ , Take[#1, 2]  {0, 0}], 0, 1]] &, #, Reverse[list]] &, {{0, 0}, {0, 1}, {1, 0}, {1, 1}}], {a_, b_, c___, a_, b_}  {b, c, a}]

Since the number of possible subsets of a set with k elements is 2 k , the number of possible real numbers is 2 ℵ 0 . … The total number of possible functions of real numbers is 2 2 ℵ 0 ; the number of continuous such functions (which can always be represented by a list of coefficients for a series) is however only 2 ℵ 0 . … Continuous cellular automata (see page 155 ) also have 2 ℵ 0 possible states.

Implementing cellular automata It is convenient to represent the state of a cellular automaton at each step by a list such as {0, 0, 1, 0, 0} , where 0 corresponds to a white cell and 1 to a black cell. … Thus, for example, rule 30 on page 27 corresponds to the list {0, 0, 0, 1, 1, 1, 1, 0} . … (In a sense there is a bug in the program in that the update only puts new values into n-2 of the n array elements.)