Rule 90R has the property that of the diamond of cells at relative positions {{-n,0},{0,-n},{n,0},{0,n}} it is always true for any n that an even number are black.

For all initial conditions this depth seems at first to increase linearly, then to decrease in a nested way according to FoldList[Plus, 0, Flatten[Table[ {1, 1, Table[-1, {IntegerExponent[i, 2] + 1}]}, {i, m}]]] This quantity alternates between value 1 at position 2 j and value j at position 2 j - j + 1 .

Game systems One can think of positions or configurations in a game as corresponding to nodes in a large network, and the possible moves in the game as corresponding to connections between nodes. … And even though the underlying rules in the game may be simple, the pattern of such winning positions is often quite complex.

At first an aggregation system (see page 331 ) might seem to be an obvious model for their growth: each new development gets added to the exterior of the city at a random position.

Random causal networks If one assumes that there are events at random positions in continuous spacetime, then one can construct an effective causal network for them by setting up connections between each event and all events in its future light cone—then deleting connections that are redundant in the sense that they just provide shortcuts to events that could otherwise be reached by following multiple connections.

Implementation [of causal networks] Given a list of successive positions of the active cell, as from Map[Last, MAEvolveList[rule, init, t]] (see page 887 ), the network can be generated using MAToNet[list_] := Module[{u, j, k}, u[_] = ∞ ; Reverse[ Table[j = list 〚 i 〛 ; k = {u[j - 1], u[j], u[j + 1]}; u[j - 1] = u[j] = u[j + 1] = i; i  k, {i, Length[list], 1, -1}]]] where nodes not yet found by explicit evolution are indicated by ∞ .

Each node is labelled by a possible position for the dot. In the first case shown, starting for example at position 4 the dot then visits positions 5, 0, 1, 2 and so on, at each step going from one node in the network to the next.

The value of the cell at position n from the end of row t is thus the n th digit of m t , or Mod[Quotient[m t , k n ], k] .

Implementation [of cellular automaton state networks] One can represent a network by a list such as {{1  2}, {0  3, 1  2}, {0  3, 1  1}} where each element represents a node whose number corresponds to the position of the element, and for each node there are rules that specify to which nodes arcs with different values lead. … = {}, 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.

Much as in both the visual and tactile systems, there seems to be a fairly direct mapping from position on the cochlea to position in the auditory cortex.