And in practice the n th digit can be found just by computing slightly over n terms of the sum, according to Round[FractionalPart[ Sum[FractionalPart[PowerMod[2, n - k, k]/k], {k, n}] + Sum[2 n - k /k, {k, n + 1, n + d}]]] where several values of d can be tried to check that the result does not change.

Only with the advent of computer graphics in the 1970s, however, does the idea appear to have arisen of varying angles to get different forms.

This can be done for blocks up to length n in a 1D cellular automaton with k colors using ReversibleQ[rule_, k_, n_] := Catch[Do[ If[Length[Union[Table[CAStep[rule, IntegerDigits[i, k, m]], {i, 0, k m - 1}]]] ≠ k m , Throw[False]], {m, n}]; True] For k = 2 , r = 1 it turns out that it suffices to test only up to n = 4 (128 out of the 256 rules fail at n = 1 , 64 at n = 2 , 44 at n = 3 and 14 at n = 4 ); for k = 2 , r = 2 it suffices to test up to n = 15 , and for k = 3 , r = 1 , up to n = 9 .

The argument suffers however from the same difficulties as the ones for chaos theory discussed in Chapter 6 and does not in the end explain in any real way the origins of randomness, or the observed validity of the Second Law.

But finally, when technology had advanced to the point where it became almost trivial for me to do so, I went back and generated some straightforward pages of pictures of all 256 elementary rules evolving from simple initial conditions.

simple programs do than in all the previous ten years put together.

For sequences involving only two distinct integers flat spectra are rare; with ± 1 those equivalent to {1, 1, 1, -1} seem to be the only examples. ( {r 2 , r s, s 2 , -r s} works for any r and s , as do all lists obtained working modulo x n - 1 from p[x]/p[1/x] where p[x] is any invertible polynomial.)

More specific criteria also included ability to use tools, plan actions, use language, solve logical problems and do arithmetic.

But this discovery does not appear to have been followed up, and systems equivalent to simple 2D Turing machines were reinvented again, largely independently, several times in the mid-1980s: by Christopher Langton in 1985 under the name "vants"; by Rudy Rucker in 1987 under the name "turmites"; and by Allen Brady in 1987 under the name "turning machines".

To do this in general one starts by picking a vector e in a timelike direction, then normalizes it to be a unit vector so that e . g . e  -1 .