Understanding nature In Greek times it was noted that simple geometrical rules could explain many features of astronomy—the most obvious being the apparent revolution of the stars and the circular shapes of the Sun and Moon.

(Note that unless one introduces an explicit copy operation—or adds variables as in the previous note—there is no way to use the same intermediate result multiple times without recomputing it.) … The following generates explicit lists of n -input Boolean functions requiring successively larger numbers of Nand operations: Map[FromDigits[#, 2] &, NestWhile[Append[#, Complement[Flatten[Table[Outer[1 - Times[##] &, # 〚 i 〛 , # 〚 -i 〛 , 1], {i, Length[#]}], 2], Flatten[#, 1]]] &, {1 - Transpose[IntegerDigits[Range[2 n ] - 1, 2, n]]}, Length[Flatten[#, 1]] < 2 2 n &], {2}] The results for 2-step cellular automaton evolution in the main text were found by a recursive procedure.

= {}) &];]] ReverseRule[a_  b_, {i_}] := {___, {s[x___, b, y___], {u___}}, ___}  {s[x, a, y], {i, u}} /; FreeQ[s[x], s[a]] In general, there will in principle be more than one such list, and to pick the appropriate list in a practical situation one normally takes the rules of the language to apply with a certain precedence—which is how, for example, x + y z comes to be interpreted in Mathematica as Plus[x, Times[y, z]] rather than Times[Plus[x, y], z] .

Position is deduced from the arrival times of signals, as determined by the relative phases of the LFSR sequences received.

A total of 40 disks were started with positions and velocities determined by a middle-square random number generator (see page 975 ), and their motion was followed for about 10 collision times—after which roundoff errors in the 64-bit numbers used had grown too big.

Corresponding to the result on page 870 for rule 90, the number of black cells at row t in the pattern from rule 150 is given by Apply[Times, Map[(2 # + 2 - (-1) # + 2 )/3 &, Cases[Split[IntegerDigits[t, 2]], k:{1 ..}  Length[k]]]] There are a total of 2 m Fibonacci[m+2] black cells in the pattern obtained up to step 2 m , implying fractal dimension Log[2, 1 + Sqrt[5]] .

The Central Limit Theorem leads to a self-similarity property for the Gaussian distribution: if one takes n numbers that follow Gaussian distributions, then their average should also follow a Gaussian distribution, though with a standard deviation that is 1/ √ n times smaller.

The repetitive structure of picture (a) implies that to reproduce this picture all we need do is to specify the colors in a 49×2 block, and then say that this block should be repeated an appropriate number of times.

But as we have seen many times in this book, more complicated rules do not necessarily produce behavior that is fundamentally any more complicated.

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.