But while this means that it might be possible for there to be arbitrariness in the causal network for the universe, it still tends to be my suspicion that there is not—and that in fact the particular rules followed by the universe do in the end have the property that they always yield the same causal network. … And just as in these systems, it is possible for intricate structures to be produced, but the structures always turn out to have a highly regular nested form.

Tag Systems One of the goals of this chapter is to find out just how simple the underlying structure of a system can be while the system as a whole is still capable of producing complex behavior. … And as a result, the pattern it produces must ultimately have a simple repetitive or nested form.

Starting with an ordinary base 2 digit sequence, one prepends a unary specification of its length, then a specification of that length specification, and so on: (Flatten[{Sign[-Range[1 - Length[#], 0]], #}] &)[ Map[Rest, IntegerDigits[Rest[Reverse[NestWhileList[ Floor[Log[2, #] &, n + 1, # > 1 &]]],2]]] (d) Binary-coded base 3. … Apply[Take, RealDigits[(N[#, N[Log[10, #] + 3]] &)[ n √ 5 /GoldenRatio 2 + 1/2], GoldenRatio]] The representations of all the first Fibonacci[n] - 1 numbers can be obtained from (the version in the main text has Rest[RotateLeft[Join[#, {0, 1}]]] & applied) Apply[Join, Map[Last, NestList[{# 〚 2 〛 ], Join[Map[Join[{1, 0}, Rest[#]] & , # 〚 2 〛 ], Map[Join[{1, 0}, #] &, # 〚 1 〛 ]]} &, {{}, {{1}}}, n-3]]]

This list can then be updated using CCAEvolveStep[f_, list_List] := Map[f, (RotateLeft[list] + list + RotateRight[list])/3] CCAEvolveList[f_, init_List, t_Integer] := NestList[CCAEvolveStep[f, #] &, init, t] where for the rule on page 157 f is FractionalPart[3#/2] & while for the rule on page 158 it is FractionalPart[# + 1/4] & .

intricate nested pattern shown below arises from many different simple programs. … But after a while it becomes clear what makes sense and what does not.

For while ordinary human language has little trouble describing repetitive and even nested patterns, it seems to be able to do very little with more complex patterns—which is in a sense why this book, for example, depends so heavily on visual presentation.

For rule 90 the combination c can be specified as {{1, 0, 1}} , while for rule 150R it can be specified as {{0, 1, 0}, {1, 1, 1}} . All generalized additive rules ultimately yield nested patterns.

means that the network of connections between stems necessarily has a very simple nested form. … Presumably most of the processes that are important take place while leaves are still folded up inside buds, and are not yet very solid.

In particular, it might be thought that the behavior of systems like rule 30—while obviously at least somewhat computationally sophisticated—might somehow be too random to be harnessed to allow complete universality. … We know from Chapter 11 that systems whose behavior is purely repetitive or purely nested cannot be universal.

Continued fractions The first n terms in the continued fraction representation for a number x can be found from the built-in Mathematica function ContinuedFraction , or from Floor[NestList[1/Mod[#, 1]&, x, n - 1]] A rational approximation to the number x can be reconstructed from the continued fraction using FromContinuedFraction or by Fold[(1/#1 + #2 )&, Last[list], Rest[Reverse[list]]] The pictures below show the digit sequences of successive iterates obtained from NestList[1/Mod[#, 1]&, x, n] for several numbers x . … Fairly large terms are sometimes seen quite early: in 5 1/3 term 19 is 3052, while in Root[10 + 8 # - # 3 &, 1] term 34 is 1,501,790. … For any irrational number this quantity cannot be less than 2, while for algebraic irrationals Klaus Roth showed in 1955 that it can only have finitely many peaks that reach above any specified level.