The discovery from the mid-1800s to the mid-1900s of all sorts of elaborate chemical processes in living systems led biologists often to view life as defined by its ability to maintain fixed overall structure while achieving chemical functions such as metabolism.

Optimal circuit blocks for operations such as addition and sorting (see page 1142 ) have occasionally been found by searches, but are more often found by explicit construction, progressive improvement or systematic logic minimization (see page 1097 ).

{x_, y_}  {{x, y}, {x + 1, y}, {x, y + 1}}, 1] &, {{0, 0}}, n] • Transpose[{Re[#], Im[#]}] &[ Flatten[Nest[{2 #, 2 # + 1, 2 # +  } &, {0}, n]]] (compare page 1005 ) • Position[Map[Split, NestList[Sort[Flatten[{#, # + 1}]] &, {0}, 2 n - 1]], _?

Without symmetry, all sorts of shapes can be obtained, as in the pictures below.

In time there will doubtless also be all sorts of additional material and educational options available.

Pointer-based encoding One can encode a list of data d by generating pointers to the longest and most recent copies of each subsequence of length at least b using PEncode[d_, b_ : 4] := Module[{i, a, u, v}, i = 2; a = {First[d]}; While[i ≤ Length[d], {u, v} = Last[Sort[Table[{MatchLength[d, i, j], j}, {j, i - 1}]]]; If[u ≥ b, AppendTo[a, p[i - v, u]]; i += u, AppendTo[a, d 〚 i 〛 ]; i++]]; a] MatchLength[d_, i_, j_] := With[{m = Length[d] - i}, Catch[ Do[If[d 〚 i + k 〛 =!

Results in the late 1900s in astrophysics and cosmology seemed to suggest that for us to exist our universe must satisfy all sorts of constraints—and to avoid explaining this in terms of purpose the Anthropic Principle was introduced (see page 1026 ).

.) • Will a given sequence of pair comparisons correctly sort any list (see page 1142 )?

Hump m in the picture of sequence (c) shown is given by FoldList[Plus, 0, Flatten[Nest[Delete[NestList[Rest, #, Length[#] - 1], 2]&, Append[Table[1, {m}], 0], m]] - 1/2] The first 2 m elements in the sequence can also be generated in terms of reordered base 2 digit sequences by FoldList[Plus, 1, Map[Last[Last[#]]&, Sort[Table[{Length[#], Apply[Plus, #], 1 - #}& [ IntegerDigits[i, 2]], {i, 2 m }]]]] Note that the positive and negative fluctuations in sequence (f) are not completely random: although the probability for individual fluctuations in each direction seems to be the same, the probability for two positive fluctuations in a row is smaller than for two negative fluctuations in a row.

But despite all sorts of elegant mathematical work, the theory remains rather distant from observed features of our universe.