Already in the early 1980s, however, my studies on simple additive and other cellular automata (see page 26 ) had for example made it rather clear that this is not the case. … To model the pouring of sand into a pile one can consider a series of cycles, in which at each cycle one first adds 4 to the value of the center cell, then repeatedly applies the rule until a new fixed configuration FixedPoint[SandStep, s] is obtained. … The behavior one sees is fairly complicated—a fact which in the past resulted in much confusion and some bizarre claims, but which in the light of the discoveries in this book no longer seems surprising.

The maximum halting times for the first few sizes n are {5, 159, 161, 1021, 5419, 315391, 1978213883, 1978213885, 3018415453261} These occur for inputs {1, 2, 5, 10, 26, 34, 106, 213, 426} and correspond to outputs (each themselves maximal for given n ) 2^{3, 23, 24, 63, 148, 1148, 91148, 91149, 3560523} - 1 Such maxima often seem to occur when the input x has the form (20 4 s - 2)/3 (and so has digits {1, 1, 0, 1, 0, … , 1, 0} ). The output f[x] in such cases is always 2 u - 1 where u = Nest[(13 + (6# + 8)(5/2)^ IntegerExponent[6# + 8, 2])/6 &, 1, s + 1] One then finds that 6u + 8 has the form Nest[If[EvenQ[#], 5#/2, # + 21]&, 14, m] for some m , suggesting a connection with the number theory systems of page 122 .

Explanations suggested for apparent absence include: • Extraterrestrials are visiting, but we do not detect them; • Extraterrestrials have visited, but not in recorded history; • Extraterrestrials choose to exist in other dimensions; • Interstellar travel is somehow infeasible; • Colonization is somehow ecologically limited; • Physical travel is not worth it; only signals are ever sent. … Often irreducible work will be required, which one might imagine it would be worthwhile to trade. … Extrapolating from our development, one might expect that most extraterrestrials would be something like immortal disembodied minds.

After a large number of steps t , the number of distinct positions visited will be proportional to t , at least above 2 dimensions (in 2D, it is proportional to t/Log[t] and in 1D √ t ). … To make a random walk on a lattice with k directions in two dimensions, one can set up e = Table[{Cos[2 π s/k], Sin[2 π s/k]}, {s, 0, k - 1}] then use FoldList[Plus, {0, 0}, Table[e 〚 Random[Integer, {1, k}] 〛 , {t}]] It turns out that on any regular lattice, in any number of dimensions, the average behavior of a random walk is always isotropic. … But in general, one can consider sources that emit new particles every step, or absorbers and reflectors of particles.

But whenever the evolution is ergodic, so that all states of a given energy are visited with equal frequency, the average behavior obtained will at least eventually correspond to the average over all states discussed above. … Of the 262,144 9-neighbor outer totalistic rules the only ones that conserve e[s] are identity and complement. … And since the evolution conserves e[s] changing the initial value of p allows one to sample different total energies.

Bell's inequalities In classical physics one can set up light waves that are linearly polarized with any given orientation. … The approach I discuss in the main text is quite different, in effect using the idea that in a network model of space there can be direct connections between particles that do not in a sense ever have to go through ordinary intermediate points in space. … And to get a kind of analog of Bell's inequalities one can look at correlations defined by such expectation values for field operators at spacelike-separated points (too close in time for light signals to get from one to another).

One issue much discussed in cosmology since the late 1970s is how the universe manages to be so uniform. … And in a case like page 518 —with spacetime always effectively having a fixed finite dimension—points that are a distance t apart tend to have common ancestors only at least t steps back. But in a case like (a) on page 514 —where spacetime has the structure of an exponentially growing tree—points a distance t apart typically have common ancestors just Log[t] steps back.

CTToR110[rules_ /; Select[rules, Mod[Length[#], 6] ≠ 0 &]  {}, init_] := Module[{g1, g2, g3, nr = 0, x1, y1, sp}, g1 = Flatten[ Map[If[#1 === {}, {{{2}}}, {{{1, 3, 5 - First[#1]}}, Table[ {4, 5 - # 〚 n 〛 }, {n, 2, Length[#]}]}] &, rules] /. a_Integer  Map[({d[# 〚 1 〛 , # 〚 2 〛 ], s[# 〚 3 〛 ]}) &, Partition[c[a], 3]], 4]; g2 = g1 = MapThread[If[#1 === #2 === {d[22, 11], s3}, {d[ 20, 8], s3}, #1] &, {g1, RotateRight[g1, 6]}]; While[Mod[ Apply[Plus, Map[# 〚 1, 2 〛 &, g2, 30] ≠ 0, nr++; g2 = Join[ g2, g1]]; y1 = g2 〚 1, 1, 2 〛 - 11; If[y1 < 0, y1 += 30]; Cases[ Last[g2] 〚 2 〛 , s[d[x_, y1], _, _, a_]  (x1 = x + Length[a])]; g3 = Fold[sadd, {d[x1, y1], {}}, g2]; sp = Ceiling[5 Length[ g3 〚 2 〛 ]/(28 nr) + 2]; {Join[Fold[sadd, {d[17, 1], {}}, Flatten[Table[{{d[sp 28 + 6, 1], s[5]}, {d[398, 1], s[5]}, { d[342, 1], s[5]}, {d[370, 1], s[5]}}, {3}], 1]] 〚 2 〛 , bg[ 4, 11]], Flatten[Join[Table[bgi, {sp 2 + 1 + 24 Length[init]}], init /. {0  init0, 1  init1}, bg[1, 9], bg[6, 60 - g2 〚 1, 1, 1 〛 + g3 〚 1, 1 〛 + If[g2 〚 1, 1, 2 〛 < g3 〚 1, 2 〛 , 8, 0]]]], g3 〚 2 〛 }] s[1] = struct[{3, 0, 1, 10, 4, 8}, 2]; s[2] = struct[{3, 0, 1, 1, 619, 15}, 2]; s[3] = struct[{3, 0, 1, 10, 4956, 18}, 2]; s[4] = struct[{0, 0, 9, 10, 4, 8}]; s[5] = struct[{5, 0, 9, 14, 1, 1}]; {c[1], c[2]} = Map[Join[{22, 11, 3, 39, 3, 1}, #] &, {{63, 12, 2, 48, 5, 4, 29, 26, 4, 43, 26, 4, 23, 3, 4, 47, 4, 4}, {87, 6, 2, 32, 2, 4, 13, 23, 4, 27, 16, 4}}]; {c[3], c[4], c[5]} = Map[Join[#, {4, 17, 22, 4, 39, 27, 4, 47, 4, 4}] &, {{17, 22, 4, 23, 24, 4, 31, 29}, {17, 22, 4, 47, 18, 4, 15, 19}, {41, 16, 4, 47, 18, 4, 15, 19}}] {init0, init1} = Map[IntegerDigits[216 (# + 432 10 49 ), 2] &, {246005560154658471735510051750569922628065067661, 1043746165489466852897089830441756550889834709645}] bgi = IntegerDigits[9976, 2] bg[s_, n_] := Array[bgi 〚 1 + Mod[# - 1, 14] 〛 &, n, s] ev[s[d[x_, y_], pl_, pr_, b_]] := Module[{r, pl1, pr1}, r = Sign[BitAnd[2^ListConvolve[{1, 2, 4}, Join[bg[pl - 2, 2], b, bg[pr, 2]]], 110]]; pl1 = (Position[r - bg[pl + 3, Length[r]], 1 | -1] /. {}  {{Length[r]}}) 〚 1, 1 〛 ; pr1 = Max[pl1, (Position[r - bg[pr + 5 - Length[r], Length[r]], 1 | -1] /. {}  {{1}}) 〚 -1, 1 〛 ]; s[d[x + pl1 - 2, y + 1], pl1 + Mod[pl + 2, 14], 1 + Mod[pr + 4, 14] + pr1 - Length[r], Take[r, {pl1, pr1}]]] struct[{x_, y_, pl_, pr_, b_, bl_}, p_Integer : 1] := Module[ {gr = s[d[x, y], pl, pr, IntegerDigits[b, 2, bl]], p2 = p + 1}, Drop[NestWhile[Append[#, ev[Last[#]]] &, {gr}, If[Rest[Last[#]] === Rest[gr], p2--]; p2 > 0 &], -1]] sadd[{d[x_, y_], b_}, {d[dx_, dy_], st_}] := Module[{x1 = dx - x, y1 = dy - y, b2, x2, y2}, While[y1 > 0, {x1, y1} += If[Length[st]  30, {8, -30}, {-2, -3}]]; b2 = First[Cases[st, s[d[x3_, -y1], pl_, _, sb_]  Join[bg[pl - x1 - x3, x1 + x3], x2 = x3 + Length[sb]; y2 = -y1; sb]]]; {d[x2, y2], Join[b, b2]}] CTToR110[{{}}, {1}] yields blocks of lengths {7204, 1873, 7088} . … The picture below shows what happens if one chops these blocks into rows and arranges these in 2D arrays. In the first two blocks, much of what one sees is just padding to prevent clock pulses on the left from hitting data in the middle too early on any given step.

The first of these points means that one is concerned with so-called pure group theory—and with finding results valid for all possible groups. … Within predicate logic one can give the relations as statements, but in effect one cannot specify that no other relations hold. … And one can now ask whether there is then universality.

Vacuum fluctuations As an analog of the uncertainty principle, one of the implications of the basic formalism of quantum theory is that an ordinary quantum field can in a sense never maintain precisely zero value, but must always show certain fluctuations—even in what one considers the vacuum. And in terms of Feynman diagrams the way this happens is by virtual particle-antiparticle pairs of all types and all energy-momenta continually forming and annihilating at all points in the vacuum. … If one has moving boundaries it turns out that vacuum fluctuations can in effect be viewed as producing real particles.