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And in this case it turns out that all patterns are in effect just simple superpositions of the basic nested pattern that is obtained by starting with a single black cell. As a result, when the initial conditions involve only a limited region of black cells, the overall pattern produced always ultimately has a simple nested form. … Any initial condition that contains black cells only in a limited region will thus lead to a pattern that ultimately has a simple nested form.
In the top four cases, the pattern produced ultimately has a simple nested form.
With two colors, it turns out that no totalistic rules yield anything other than repetitive or nested behavior. … Allowing four or more colors, however, does not further increase the complexity of the behavior, and, as the picture shows, even with five colors, simple repetitive and nested behavior can still occur.
But at an overall level, all the patterns have exactly the same basic nested structure. … All these patterns ultimately have the same overall nested form.
If one restricts oneself to a fixed number of elementary mathematical Nested patterns constructed using arithmetic operations. … The limiting arrangements of colors correspond to nested patterns.
But once in every few thousand rules, one sees behavior of the kind shown below—that is not purely repetitive, but instead has a kind of nested structure. A mobile automaton with slightly more complicated rules that yields a nested pattern.
In every example in this book where nested patterns like those from rule 90 are obtained it turns out that the underlying rules that are responsible can be set up to behave exactly like rule 90. … Why does it yield nested patterns? … One consequence of this is that the patterns produced by rule 90 have a nested or self-similar form.
So what about nested patterns? … And with this idea the picture on the next page shows an example of how what is in effect a formula can be constructed for a nested pattern. … As we have discussed several times in this book, any nested pattern must—almost by definition—be able to be reproduced by a neighbor-independent substitution system.
Sierpiński pattern Other ways to generate step n of the pattern shown here in various orientations include: • Mod[Array[Binomial, {2, 2} n , 0], 2] (see pages 611 and 870 ) • 1 - Sign[Array[BitAnd, {2, 2} n , 0]] (see pages 608 and 871 ) • NestList[Mod[RotateLeft[#] + #, 2] &, PadLeft[{1}, 2 n ], 2 n - 1] (see page 870 ) • NestList[Mod[ListConvolve[{1, 1}, #, -1], 2] &, PadLeft[{1}, 2 n ], 2 n - 1] (see page 870 ) • IntegerDigits[NestList[BitXor[2#, #] &, 1, 2 n - 1], 2, 2 n ] (see page 906 ) • NestList[Mod[Rest[FoldList[Plus, 0, #]], 2] &, Table[1, {2 n }], 2 n - 1] (see page 1034 ) • Table[PadRight[ Mod[CoefficientList[(1 + x) t - 1 , x], 2], 2 n - 1], {t, 2 n }] (see pages 870 and 951 ) • Reverse[Mod[CoefficientList[Series[1/(1 - (1 + x)y), {x, 0, 2 n - 1}, {y, 0, 2 n - 1}], {x, y}], 2]] (see page 1091 ) • Nest[Apply[Join, MapThread[ Join, {{#, #}, {0 #, #}}, 2]] &, {{1}}, n] (compare page 1073 ) The positions of black squares can be found from: • Nest[Flatten[2# /. {x_, y_}  {{x, y}, {x + 1, y}, {x, y + 1}}, 1] &, {{0, 0}}, n] • Transpose[{Re[#], Im[#]}] &[ Flatten[Nest[{2 #, 2 # + 1, 2 # +  } &, {0}, n]]] (compare page 1005 ) • Position[Map[Split, NestList[Sort[Flatten[{#, # + 1}]] &, {0}, 2 n - 1]], _?(OddQ[Length[#]] &), {2}] (see page 358 ) • Flatten[Table[Map[{t, #} &, Fold[Flatten[{#1, #1 + #2}] &, 0, Flatten[2^(Position[ Reverse[IntegerDigits[t, 2]], 1] - 1)]]], {t, 2 n - 1}], 1] (see page 870 ) • Map[Map[FromDigits[#, 2] &, Transpose[Partition[#, 2]]] &, Position[Nest[{{#, #}, {#}} &, 1, n], 1] - 1] (see page 509 ) A formatting hack giving the same visual pattern is DisplayForm[Nest[SubsuperscriptBox[#, #, #] &, "1", n]]
In general, there is great variety in the possible structures that can be set up in network systems, and as one further example the picture below shows a network that forms a nested pattern. … And although this representation can obscure the geometrical structure An example of a network that forms a nested geometrical structure.
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