The distributive laws appear at positions 2813 and 2814 in the list; it takes a long proof to obtain the second one from preceding theorems.

The result of this is that any change in the initial position of a point will be amplified by a factor of two at each step. And while a particular machine may be able to control the initial position of a point to a certain accuracy, such repeated amplification will eventually lead to sensitivity to still smaller changes.

In an array of cells like in a cellular automaton each cell is always assigned some definite position. But in a network of nodes, the nodes are not intrinsically assigned any position.

But now what the cellular automata do is to take specifications of positions of cells, and then in effect compute directly from these the colors of cells. The way things are set up the initial conditions for these cellular automata consist of digit sequences of numbers that give positions.

[Causal networks for] 2D mobile automata As in 2D random walks, active cells in 2D mobile automata often do not return to positions they have visited before, with the result that no causal connections end up being created.

As another example, the Global Positioning System (GPS) works by having 24 satellites each transmit maximal length sequences from different length 10 LFSRs. Position is deduced from the arrival times of signals, as determined by the relative phases of the LFSR sequences received.

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]]

This particular structure is fairly simple: it just remains fixed in position and repeats every two steps. … It turns out, however, that initial condition 189 suddenly yields a much simpler structure—that just stays unchanged in one position at every step.

And in ordinary space, this is normally calculated by subtracting the numerical coordinates of the positions of the points. … And a test is to see whether there is a way to lay out the nodes in the network in ordinary space so that the distances between nodes computed from their positions in space agree—at least in some approximation—with the distances computed directly by following connections in the network.

And in fact the basic notion of extending the idea of position in space to an idea of position in time has been common in scientific thought for more than five centuries.