Chapter 4: Systems Based On Numbers
the notion of numbers p115.
short, non-tech, 1 page, intro on using positional notation to represent numbers as symple systems. Says even simple systems based on number can have complex behavior, and how traditional math is misleading in such discovery.
In this chapter, Wolfram uses number systems to advance his view that simple system can have complex behavior. He bases his study mostly by representing numbers using the positional notation system (⁖ binary, decimal) in a visual equivalent, and uses many classical problems in number theory, as well as some recursion, to advance his thesis.
However, the premise that number systems (as represented by positional notation systems) is a simple system, is not correct. The most simple system to represent numbers, would be counting. That is, 3 is just 3 cells, 10 is 10 cells. The positional notation system is developed much later in the history of mathematics. The positional notation system is rather complex, fundamentally relies on powers (⁖ n^0, n^1, n^2, n^3 etc for base n). So, it is not really surprising, that simple repeated arithemetic operations, can create complex behavior with respect to the digits. In a sence, the positional notation system for numbers can be thought of as a transformation rule. We could invent a simple system of transformation rule to any CA, applying to each row. When this transformation rule is applied to represent any CA, then it is basically another arbitirary rule, so that even those uniteresting repeatitive CA might now have a complex behavior when “represented” thru such transformation rule.
later, on p142, wolfram seems to address this problem by representing numbers as continued fractions. Note however, that continued fractions is like a transformation rule too whose outcome is non-trivial.
elementary arithmetic p117
p117 write number in base 2. White white square to represent 0 and black to represent 1, one number per row, stack the next number after it, right aligned. The sequence 1,2,3… will show a nested pattern. Other arithemetic sequences also shows similar pattern.
the fact that nesting behavior is shown when a arithemetic sequence is represented as a CA-like visualisation based on their positional notation system, seems inherent and trivial. It would be interesting to see such visualization of all arithemitc sequences and number bases, to see what nesting patten they actually create. It is likely that they are all binary trees.
p120. use a geometric sequence. Each term is show as binary number in a row of cells. A pict of 1,3,9,27 … is shown; it seems random with many recognizable black or white triangles, like those basic 1D CA with random triangles.
take a sequence starting with 1 and multiply by 3/2 successively. Now plot the fractional parts of each number in the sequence. consider the length of the fractial part. A pict is shown in p122. Says that this is apparently random.
p122 create a sequence: start with 1, then If[EvenQ[n], 3/2*n, 3/2*(n+1)]. Then, plot this sequence in binary as before. The pattern is random with different sized black or white small triangles. Each number in the sequence may be odd or even. It is asserted that this is even/odd aspect is “completely random”.
p123 Consider a sequence with this function f[n+1]:= If[EvenQ@n, 5/2*n, (n+1)/2]. Then, create several such sequences each with different starting number. Of the starting numbers from 1 to 10, 6 and 8 shows randomish pattern. Take the sequece that starts with 6 to a million terms. The number will have 48544 base 10 digits. Still, the pattern seems random.
Compare the number system and the CA system. Both exhibit random patterns. The CA case the pattern is generated directely based on previous cells. However, in the number based patterns there's no such locality. It is assert thus, that details of the underlying system “…do not seem to have a crucial effect on the kinds of behavior that can occur.” p125.
p125 Start with a number: n. Write it in base 2. take this number and reverse the digits then add to n. The patterns formed this way are remarkably similar to the basic 1D CA with random triangles. In the case of start number 512, at step one million it still seems random. (in the case of starting with 16, it settels to repetition after 180 steps.)
recursive sequences p128
p128 here we investigate sequences generated by having nth term depends on one or more of its previous terms. ⁖ fibonacii sequence.
most such sequence refers to previous terms in a fixed distance. Here we also investigate the previous term we refers to depends on another previous term. ⁖ f[n]:=f[n- f[n-1]]. Here, one must be careful to be sure that the referred sequence does not go before 1.
the vast majority of such sequence randomly generate will have problems of refering to senseless terms such as f[-5]. However, not all. p129, 8 cases are shown. ⁖ case h: f[n]:=f[f[n-1]]+f[n-2*6[n-1]+1], f=1, f=1.
These sequenes are plotted in a standard way. It is shown in some cases that they fluctuate randomly. The assertion in this section is that even systems based on simple addition and substraction “are capable of creating behaviors of great complexity”.
This section is somewhat interesting, in that it studies sequences that refers to previous sequencs that refers to previous sequences. (a nested reference) Off hand, i do not think there is any math that ever uses such a sequence.
according to the text, vast majority of such sequences will refer to non-existant terms.
the book shows a few such sequence (that doesn't refer to non-existant terms). Plot a sequence's graph, and also plot a sequence obtained as the diff of adjacent terms. The plots of the difference sequence seems bewilding, because some of them shows wild fluctiation.
it's unclear what can be concluded from these select examples of nested reference sequences. Wolfram uses the difference sequence to push the idea of simples programs producing complexity.
Note, the usual visual pattern of CA is not used to represent results in this section. Also, no graph about partial sums of the these sequences are shown. Nor, product are discussed. Also, the examples of recursive sequences involves only plus and minus. It might be interesting also to check out those involving multiplication. Also, the book only shows like 15 or so examples. It does not give a overall picture what such sequences are like, or some general remark. (other than that vast majority of them refers to non-existant terms) Also, it'd be interesting to see if there are any applications of nested recursive sequence.
To solve the problem of refering to non-existant terms, one could use a cyclic counting of terms.
the sequence of primes
p132. Presenting the sieve of Eratosthenes algorithm to produces primes in a visual, CA-like way. Then, graphs in various ways of prime distribution (⁖ the difference between adjacent primes). Says that primes has complex behavior. Says that traditional study on primes tried to look for regularity and failed to recognize the concept of simple programs producing complexity.
p135. Plots other simple number properties, such as number of divisors of a number, sum of divisors, expressing number as sum of 3 squares, etc. Says that primes is not unique in the sense of simple program that produce complexity.
mathematical constants p136
Uses digit of π as evidence of complexity inherent in most basic definitions.
p140. Uses digits of various square roots to claim that for practical purposes they are random.
p141. gives a procedure for generating digit sequences of square roots.
p142. shows the decimal and binary digits of 3rd and 4th roots, log of base 2, 3, and e and e^2, and claims that these again “seem for practical purposes random”.
p143. Gives a uniform formal representation of numbers in Positional notation and Continued fractions form. Namely, one involves nested addition and multiplication, and the other is nested addition and division. .
p145. Plots some simple combination of functions such as Sin[x]+Sin[Sqrt * x] + Sin[Sqrt *x], and others such as Riemann zeta function. Says that these has complex behavior.
p147. Shows how some sum of Sin or Cosine functions corrseponds to his “generalized substitution system”.