One of the defining features of space as we normally experience it is a certain locality that leads most things that happen at some particular position to be able at first to affect only things very near them.

The rule is set up so that if the value of n is written in the form i + 5 , 2 a , 3 b then the values of i , a and b on successive steps correspond respectively to the position of the register machine in its program, and to the values of the two registers (2 and 3 appear because they are the first two primes; 5 appears because it is the length of the register machine program).

And in addition, the rule for a Turing machine can depend on the state of the head, and on the color of the cell at the position of the head, but not on the colors of any neighboring cells.

The local positions of stars are generally assumed to be random. 88 constellations are usually named—quite a few presumably already identified by the Babylonians and Sumerians around 2000 BC.

Given an original DNF list s , this can be done using PI[s, n] : PI[s_, n_] := Union[Flatten[ FixedPointList[f[Last[#], n] &, {{}, s}] 〚 All, 1 〛 , 1]] g[a_, b_] := With[{i = Position[Transpose[{a, b}], {0,1}]}, If[Length[i]  1 && Delete[a, i] === Delete[b, i], {ReplacePart[a, _, i]}, {}]] f[s_, n_] := With[ {w = Flatten[Apply[Outer[g, #1, #2, 1] &, Partition[Table[ Select[s, Count[#, 1]  i &], {i, 0, n}], 2, 1], {1}], 3]}, {Complement[s, w, SameTest  MatchQ], w}] The minimal DNF then consists of a collection of these prime implicants. … Given the original list s and the complete prime implicant list p the so-called Quine–McCluskey procedure can be used to find a minimal list of prime implicants, and thus a minimal DNF: QM[s_, p_] := First[Sort[Map[p 〚 # 〛 &, h[{}, Range[Length[s]], Outer[MatchQ, s, p, 1]]]]] h[i_, r_, t_] := Flatten[Map[h[Join[i, r 〚 # 〛 ], Drop[r, #], Delete[Drop[t, {}, #], Position[t 〚 All, # 〛 ], {True}]]] &, First[Sort[Position[#, True] &, t]]]], 1] h[i_, _, {}] := {i} The number of steps required in this procedure can increase exponentially with the length of p .

Cellular automaton rules as formulas The value a[t, i] for a cell on step t at position i in any of the cellular automata in this chapter can be obtained from the definition a[t_, i_] := f[a[t - 1, i - 1], a[t - 1, i], a[t - 1, i + 1]] Different rules correspond to different choices of the function f . For example, rule 90 on page 25 corresponds to f[1, _, 1] = 0; f[0, _, 1] = 1; f[1, _, 0] =1; f[0, _, 0] = 0 One can specify initial conditions for example by a[0, 0] = 1; a[0, _] = 0 (the cell on step 0 at position 0 has value 1, but all other cells on that step have value 0). Then just asking for a[4, 0] one will immediately get the value after 4 steps of the cell at position 0.

In 3 dimensions, the positions of black cells can conveniently be displayed using Graphics3D[Map[Cuboid[-Reverse[#]] &, Position[a, 1]]]

Among k = 2 , r = 1 elementary cellular automata it turns out that this happens precisely for those 30 rules that are additive with respect to at least the first or last position on which they depend (see pages 601 and 1087 ); this includes both rules 90 and 150 and rules 30 and 45. … Out of these 141,884 are onto—and 11,388 of these turn out not to be additive with respect to any position.

[Networks generated by] random replacements As indicated in the note above, applying the second rule (T1, shown as (b) on page 511 ) at an appropriate sequence of positions can transform one planar network into any other with the same number of nodes. The pictures below show what happens if this rule is repeatedly applied at random positions in a network.

In 1918 Hermann Weyl tried to reproduce electromagnetism by adding the notion of an arbitrary scale or gauge to the metric of general relativity (see page 1028 )—and noted the "gauge invariance" of his theory under simultaneous transformation of electromagnetic potentials and multiplication of the metric by a position-dependent factor. Following the introduction of the Schrödinger equation in quantum mechanics in 1926 it was almost immediately noticed that the equations for a charged particle in an electromagnetic field were invariant under gauge transformations in which the wave function was multiplied by a position-dependent phase factor.