{{x_}  s[x], {x_, y_}  p[x, y]}), {_r, s[v : (0 | 1)], _}  r[v], {_r, b, _}  m[b], {s[0 | 1], m[b], _}  b, {_, v_, _}  v} where specific values for cells can be obtained from {b  0, s[0]  1, m[0]  2, p[0, 0]  3, r[0]  4, p[0, 1]  5, p[1, 0]  6, r[1]  7, p[1, 1]  8, m[1]  9, m[b]  10, s[1]  11} An initial condition consisting of a single element with color i in the substitution system is represented by m[i] surrounded by b 's in the cellular automaton. The specific definition given above works for neighbor-independent substitution systems whose elements have two possible colors, and in which each element is replaced at each step by at most two new elements.

In case (b), with rows chosen to be 2 j elements in length, the leftmost column will always be identical to the beginning of the sequence, and in addition every interior element will be black exactly when the cell at the top of its column has the same color as the one at the beginning of its row.

In each case the n th element appears at coordinates Sqrt[n] {Cos[n θ],Sin[n θ]} .

This mechanism can work in several ways; typically it will involve a rotary element that determines which case of the rule to use at each step.

The scheme for numbering rules works so that if the value of a particular cell is q , the value of its left neighbor is p , and the value of its right neighbor is r , then the element at position 8 - (r + 2(q + 2p)) in the list obtained from ElementaryRule will give the new value of the cell. … The result is that a list is produced which specifies for each cell which element of the rule applies to that cell.

Uniformity in space can be achieved almost trivially if each element in a system independently evolves to the same state.

In human visual perception the color of something tends to seem different depending on what is around it—so that for example a red element tends to look purple or pink if the elements around it are respectively blue or white.

One-element-dependence tag systems [emulating TMs] Writing the rule {3, {{0, _, _}  {0, 0}, {1, _, _}  {1, 1, 0, 1}}} from page 895 as {3, {0  {0, 0}, 1  {1, 1, 0, 1}}} the evolution of a tag system that depends only on its first element is obtained from TS1EvolveList[rule_, init_, t_] := NestList[TS1Step[rule, #] &, init, t] TS1Step[{n_, subs_}, {}] = {} TS1Step[{n_, subs_}, list_] := Drop[Join[list, First[list] /. subs], n] Given a Turing machine in the form used on page 888 the following will construct a tag system that emulates it: TMToTS1[rules_] := {2, Union[Flatten[rules /.

Sets are called recursive if there is a recursive function that can test whether or not any given element is in them. Sets are called recursively enumerable if there is a recursive function that can eventually generate any element in them.

Brainteasers In many puzzles and IQ tests the setup is to give a few elements in some sequence of numbers, strings or pictures, then to ask what the next element would be.