Chaos Theory and Randomness from Initial Conditions…[No text on this page] An arrangement of mirrors set up to exhibit randomness arising from sensitive dependence on initial conditions. … 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.

For all one ever need do is to work out the remainder from dividing the position of a particular square by the size of the basic repeating block, and this then immediately tells one how to look up the color one wants. … And then in the specific case shown one compares corresponding digits in these two sequences, and if these digits are ever respectively 0 and 1, then the square is white; otherwise it is black. … So as the picture illustrates this means that new squares always have positions that involve numbers containing one extra digit.

[No text on this page] Examples of patterns set up so that a short computation can be used to determine the color of each cell from the numbers representing its position. … In every picture both x and y run from 1 to 127. d[n] stands for IntegerDigits[n, 2, 7] . (h) is equivalent to digit sequences of powers of 3 in base 2 (see page 120 ).

that a digit which appears at a particular position in their result can depend on digits that were originally far away from it. … An example of a system defined by the following rule: at each step, take the number obtained at that step and write its base 2 digits in reverse order, then add the resulting number to the original one.

[No text on this page] Examples of the evolution of two-dimensional cellular automata with various totalistic rules starting from random initial conditions. … Each successive base 2 digit in the code number for the rule gives the outcome when the total of the cell and its four neighbors runs from 5 down to 0.

Digits of pi The digits of π shown here can be obtained in less than a second from Mathematica on a typical current computer using N[ π , 7000] . Historically, the number of decimal digits of π that have been computed is roughly as follows: 2000 BC (Babylonians, Egyptians): 2 digits; 200 BC ( Archimedes ): 5 digits; 1430 AD: 14 digits; 1610: 35 digits; 1706: 100 digits; 1844: 200 digits; 1855: 500 digits; 1949 (ENIAC computer): 2037 digits; 1961: 100,000 digits (IBM 7090); 1973: 1 million; 1983: 16 million; 1989: 1 billion; 1997: 50 billion; 1999: 206 billion. In the first 200 billion digits, the frequencies of 0 through 9 differ from 20 billion by {30841, -85289, 136978, 69393, -78309, -82947, -118485, -32406, 291044, -130820} An early approximation to π was 4 Sum[(-1) k /(2k + 1), {k, 0, m}] 30 digits were obtained with 2 Apply[Times, 2/Rest[NestList[Sqrt[2 + #]&, 0, m]]] An efficient way to compute π to n digits of precision is (# 〚 2 〛 2 /# 〚 3 〛 )& [NestWhile[Apply[Function[{a, b, c, d}, {(a + b)/2, Sqrt[a b], c - d (a - b) 2 , 2 d}], #]&, {1, 1/Sqrt[N[2, n]], 1/4, 1/4}, # 〚 2 〛 ≠ # 〚 2 〛 &]] This requires about Log[2, n] steps, or a total of roughly n Log[n] 2 operations (see page 1134 ).

This is very different from what happens in cases (a) and (b). … But as soon as one looks at digit sequences, it immediately becomes much clearer. … And for a while these digits are

Given that a branch with a certain color has been reached, the color of the branch to be taken next is then determined purely by the next digit in the digit sequence of n . … Note that if the rule for the finite automaton is represented for example as {{1, 2}, {2, 1}} where each sublist corresponds to a particular state, and the elements of the sublist give the successor states with inputs Range[0, k - 1] , then the n th element in the output sequence can be obtained from Fold[rule 〚 #1, #2 〛 &, 1, IntegerDigits[n - 1, k] + 1] - 1 while the first k m elements can be obtained from Nest[Flatten[rule 〚 # 〛 ] &, 1, m] - 1 To treat examples such as case (c) where elements can subdivide into blocks of several different lengths one must generalize the notion of digit sequences. In base k a number is constructed from a digit sequence a[r] , … , a[1] , a[0] (with 0 ≤ a[i] < k ) according to Sum[a[i] k i , {i, 0, r}] .

Digit sequence encryption One can consider using as encrypting sequences the digit sequences of numbers obtained from standard mathematical functions. As discussed on page 139 such digit sequences often seem locally very random. … Thus, for example, given the digit sequence of √ s one can retrieve the key s just by squaring the number obtained from early digits in the sequence.

[Patterns from] arbitrary digit operations If the operation on digit sequences that determines whether a square will be black can be performed by a finite automaton (see page 957 ) then the pattern generated must always be either repetitive or nested. … Scanning the digit sequences from the left, one starts with 0 open parentheses, then adds 1 whenever corresponding digits in the x and y coordinates differ, and subtracts 1 whenever they are the same. … Picture (b) has a black square wherever digits at more than half the possible positions differ between the x and y coordinates.