Averaged over all nodes, however, the number of nodes at distance up to r approximates r^Log[2, 3] , implying an effective dimension of Log[2, 3] .

Geodesics On a sphere all geodesics are arcs of great circles.

Starting from the set of all possible sequences, as given by AllNet[k_:2] := {Thread[(Range[k] - 1)  1]} this then yields for rule 126 the network {{0  1, 1  2}, {1  3, 1  4}, {1  1, 1  2}, {1  3, 0  4}} It is always possible to find a minimal network that represents a set of sequences. … = {}, AllNet[k], q = ISets[b = Map[Table[ Position[d, NetStep[net, #, a]] 〚 1, 1 〛 , {a, 0, k - 1}]&, d]]; DeleteCases[MapIndexed[#2 〚 2 〛 - 1  #1 &, Rest[ Map[Position[q, #] 〚 1, 1 〛 &, Transpose[Map[Part[#, Map[ First, q]]&, Transpose[b]]], {2}]] - 1, {2}], _  0, {2}]]] DSets[net_, k_:2] := FixedPoint[Union[Flatten[Map[Table[NetStep[net, #, a], {a, 0, k - 1}]&, #], 1]]&, {Range[Length[net]]}] ISets[list_] := FixedPoint[Function[g, Flatten[Map[ Map[Last, Split[Sort[Part[Transpose[{Map[Position[g, #] 〚 1, 1 〛 &, list, {2}], Range[Length[list]]}], #]], First[#1]  First[#2]&], {2}]&, g], 1]], {{1}, Range[2, Length[list]]}] If net has q nodes, then in general MinNet[net] can have as many as 2 q -1 nodes.

If one considers rules that depend on 4 rather than 3 cells, then the results turn out to be surprisingly similar: out of all 65536 possible such rules the ones with longest periods essentially always seem to be variants of rules 45, 30 or 60. … For size 15, the 33 rules with the longest periods are all additive with respect to one position.

For it shows that all sorts of sophisticated characteristics can emerge from the very same kinds of simple components. … In addition, the Principle of Computational Equivalence implies that all sorts of systems in nature and elsewhere will inevitably exhibit features that in the past have been considered unique to intelligence—and this has consequences for the mind-body problem, the question of free will, and recognition of other minds.

History [of randomness] In antiquity, it was often assumed that all events must be governed by deterministic fate—with any apparent randomness being the result of arbitrariness on the part of the gods. Around 330 BC Aristotle mentioned that instead randomness might just be associated with coincidences outside whatever system one is looking at, while around 300 BC Epicurus suggested that there might be randomness continually injected into the motion of all atoms.

There are 32 possible symmetric such rules with just 4 immediate neighbors—of which 16 lead to growth (from any seed), and all seem to yield at least approximately circular clusters (of varying densities). Without symmetry, all sorts of shapes can be obtained, as in the pictures below.

Indeed, all that it shows is that if there is complexity in the details of the initial conditions, then this complexity will eventually appear in the large-scale behavior of the system.

So now a question that arises is whether all the complexity we have seen in the past three chapters [ 2 , 3 , 4 ] somehow depends on the discreteness of the elements in the systems we have looked at.

With all the work that was done on continuous systems in the history of traditional science and mathematics, there were undoubtedly many cases in which effects related to the phenomenon of complexity were seen.