The initial condition for the system is specified by the position of the incoming light ray in the gray region at the top of each picture.

A plot of the position of a ball rolled with various initial speeds. … The ball is half white and half black, and the stripes in the picture indicate which color is on top when the ball is at a particular position.

For size 15, the 33 rules with the longest periods are all additive with respect to one position. The pictures below show the first rules that are not additive with respect to any position. … The first two are additive with respect to one position, but do not appear to be directly related to rules 45 or 30; the last two are not additive with respect to any position.

In the expanded tag system evolution, successive colors of elements are encoded by having a black cell at successive positions inside a fixed block of white cells.

Implementation of generalized mobile automata The state of a generalized mobile automaton at a particular step can be specified by {list, nlist} , where list gives the values of the cells, and nlist is a list of the positions of active cells. The rule can be given by specifying a list of cases such as {0, 0, 0}  {1, {1, -1}} , where in each case the second sublist specifies the new relative positions of active cells.

And thus for example in picture (a) there are downward connections that never reach any other node—reflecting the presence of positions on the left in the mobile automata evolution to which the active cell never returns.

Of course, to make a picture of a network system, one has to choose particular positions for each of its nodes. But the crucial point is that these positions have no fundamental significance: they are introduced solely for the purpose of visual representation.

Numbering scheme [for 2D constraints] The constraint numbered n allows the templates at Position[IntegerDigits[n, 2, 32], 1] in the list below.

Implementation [of repetitive array] The color of a cell at position {x, y} in the pattern shown is given by Extract[{{1, 0, 1}, {0, 1, 0}}, Mod[{y, x}, {2, 3}] + 1] .

(b) (Thue–Morse sequence) The color s[n] of the element at position n is given by 1 - Mod[DigitCount[n - 1, 2, 1], 2] . … The color of the element at position n is given by 2 - (Floor[(n + 1) GoldenRatio] - Floor[n GoldenRatio]) (see page 904 ), while the position of the k th white element is given by the so-called Beatty sequence Floor[k GoldenRatio] . … The picture below shows the number of black cells that occur before position n .