Maximal block compression If one has data that consists of a long sequence of blocks, each of length b , and each independently chosen with probability p[i] to be of type i , then as argued by Claude Shannon in the late 1940s, it turns out that the minimum number of base 2 bits needed on average to represent each block in such a sequence is h = -Sum[p[i] Log[2, p[i]], {i, 2 b }] . … If only one block occurs with nonzero probability then h  0 . … For p[i] that are not powers of 1/2, non-integer length codewords would be required.

If 2r is less than q , the digit generated at that step is 0, and r is replaced by 2r . Otherwise, r is replaced by 2r – q . … Perfect squares such as 4 = 2 × 2 and 9 = 3 × 3 are specifically set up to have square roots that are just whole numbers.

Substitution systems [from cellular automata] Given a substitution system with rules in the form such as {1  {0}, 0  {0, 1}} used on page 889 , the rules for a cellular automaton which emulates it are obtained from SSToCA[rules_] := {{b, b, p[x_, _]}  s[x], {_, s[v : (0 | 1)], p[x_, _]}  p[v, x], {_, p[_, y_], _}  s[y], {_, s[v : (0 | 1)], _m}  m[v], {s[0 | 1], m[v : (0 | 1)], _}  s[v], {b, m[v : (0 | 1)], _}  r[v], {_, r[v : (0 | 1)], _}  (Replace[v, rules] /. {{x_}  s[x], {x_, y_}  p[x, y]}), {_r, s[v : (0 | 1)], _}  r[v], {_r, b, _}  m[b], {s[0 | 1], m[b], _}  b, {_, v_, _}  v} where specific values for cells can be obtained from {b  0, s[0]  1, m[0]  2, p[0, 0]  3, r[0]  4, p[0, 1]  5, p[1, 0]  6, r[1]  7, p[1, 1]  8, m[1]  9, m[b]  10, s[1]  11} An initial condition consisting of a single element with color i in the substitution system is represented by m[i] surrounded by b 's in the cellular automaton.

If no self connections are allowed then these numbers become {1, 2, 6, 20, 91} , while if neither self nor multiple connections are allowed (yielding what are often referred to as cubic or 3-regular graphs), the numbers become {0, 1, 2, 5, 19, 85, 509, 4060, 41301, 510489} , or asymptotically (6 n)! … (2 n)! … If one requires the networks to be planar the numbers are {0, 1, 1, 3, 9, 32, 133, 681, 3893, 24809, 169206} .

And following my original work on cellular automata these choices can be numbered from 0 to 255, as in the second picture below. … As indicated, the rules can conveniently be numbered from 0 to 255. The number assigned is such that when written in base 2, it gives a sequence of 0's and 1's that correspond to the sequence of new colors chosen for each of the eight possible cases covered by the rule.

Implementation [of texture perception model] The exact matches for a template σ in data containing elements 0 and 1 can be obtained from Sign[ListCorrelate[2 σ - 1, data] - Count[ σ , 1, 2]] + 1

Comments on Mathematica functions CenterList works by first creating a list of n 0's, then replacing the middle 0 by a 1. (In Mathematica 4 and above PadLeft[{1}, n, 0, Floor[n/2]] can be used instead.) … The actual list of new cell values is then generated by using the fact that {i, j, k} 〚 {2, 1, 1, 3, 2} 〛 ⟶ {j, i, i, k, j} .

But for Sin[a x] + Sin[b x] + Sin[c x] , there are usually zeros off the real axis (even say for a = 1 , b = 3/2 , c = 5/3 ), as shown in the pictures below. … But in a case like Sin[x] + Sin[ √ 2 x] + Sin[ √ 3 x] there is a continuous distribution of spacings between zeros, as shown on a logarithmic scale below. (For 0 < x < 10 6 there are a total of 448,494 zeros, with maximum spacing ≃ 4.6 and minimum spacing ≃ 0.013 .)

The number of black cells on row t is given by 2^DigitCount[t, 2, 1] , where DigitCount[t, 2, 1] is plotted on page 902 . 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] . … In this pattern, the color of a particular cell can be obtained directly from the digit sequences for t and n by 1 - Sign[BitAnd[-t, n]] or (see page 583 ) With[{d = Ceiling[Log[2, Max[t, n] + 1]]}, If[FreeQ[ IntegerDigits[t, 2, d] - IntegerDigits[n, 2, d], -1], 1, 0]]

Whether the light ray goes to the left or to the right at each step is then determined by successive digits in the base 2 representation for the number that gives the initial condition. … The initial condition used here is π/4 , which has digit sequence 0.1100100100001111111.