Packing deformable objects If one pushes together identical deformable objects in 2D they tend to arrange themselves in a regular hexagonal array—and this configuration is known to minimize total boundary length. … Note that for a 3D Voronoi diagram with randomly placed points, the average number of faces for each region is 2 + 48 π 2 /35 ≃ 15.5 .) … But in 1993 Denis Weaire and Robert Phelan discovered a layered repetitive arrangement of 12- and 14-faced polyhedra (average 13.5) that yields 0.003 times less total area.

For rule 184, it can be taken to be 1 for each block, and to be 0 otherwise. … For rule 150, it is 1 for and , with all computations done modulo 2. In general, if the global conserved quantity involves blocks of size b , the flux can be computed by looking at blocks of size b + 2r - 1 .

More colors [in additive cellular automata] The pictures below show generalizations of rule 90 to k possible colors using the rule CAStep[k_Integer, a_List] := Mod[RotateLeft[a] + RotateRight[a], k] or equivalently Mod[ListCorrelate[{1, 0, 1}, a, 2], k] . … For prime k the total number of non-white cells down to step k m is (1/2k (k + 1)) m and the patterns have fractal dimension 1 + Log[k, (k + 1)/2] (see page 955 ).

And if there are going to be an infinite number of excluded blocks, there must be additional excluded blocks with lengths between n and 2n . In rule 126, the lengths of the shortest newly excluded blocks on successive steps are 0, 3, 12, 13, 14, 14, 17, 15. It is common to see such lengths progressively increase, although in principle they can decrease by as much as 2r from one step to the next.

Turing machines [from cellular automata] Given any Turing machine with rules in the form used on page 888 and k possible colors for each cell, a cellular automaton which emulates it can be constructed using TMToCA[rules_, k_:2] := Flatten[{Map[g[#, k]&, rules], {_, x_, _}  x}] g[{s_, a_}  {sp_, ap_, d_}, k_] := {If[d  1, Identity, Reverse][{k s + a, x_, _}]  k sp + x, {_, k s + a, _}  ap} If the Turing machine has s states for its head, then the cellular automaton has k (s+1) colors for each cell. An initial condition with a single cell of color k surrounded by 0's corresponds to being in state 1 with a blank tape in the Turing machine.

(For Xor , there are 2 possible σ , while for Or there are 3.) … And the only way to achieve this is to have a rule in which the new value of a cell is given by a linear form such as a x + b y . … If σ [x] is required to be continuous, then the only form it can have is c x .

Nonlinear feedback shift registers Linear feedback shift registers of the kind discussed on page 974 can be generalized to allow any function f (note the slight analogy with cyclic tag systems): NLFSRStep[f_, taps_, list_] := Append[Rest[list], f[list 〚 taps 〛 ]] With the choice f=IntegerDigits[s, 2, 8] 〚 8 - # . {4, 2, 1} 〛 & and taps = {1, 2, 3} this is essentially a rule s elementary cellular automaton. … One set of computations concerned functions f[{w_, x_, y_, z_}] := Mod[w + y + z + x y + x z + y z, 2] (apparently chosen to have balance between 0's and 1's that would minimize correlations). … And as noted by Nicolaas de Bruijn in 1946 there are 2 2 n - 1 -n such paths with length 2 n , and thus this number of functions f out of the 2 2 n possible must yield sequences of maximal length.

Finding densest packings of n circles is in general like solving quadratic programming problems with about n 2 constraints. … This will not happen, however, for size ratios ≤ 2/ √ 3 - 1 ≃ 0.15 , since then the small circles can fit into the interstices of an ordinary hexagonal pattern, yielding a filling fraction 1/18(17 √ 13 - 24) π ≃ 0.95 . … (The total numbers of different sizes of circles for the first few steps are {2, 3, 5, 10, 24, 63, 178, 521} .

A simple cellular automaton whose evolution effectively computes the remainder after division of a number by 2. Starting from a row of n black cells, 0 black cells survive if n is even, and 1 black cell survives if n is odd.

(In case (e), for example, the equation is t n  -t n - 1 + t n - 2 .) … A recurrence relation like f[0] = x; f[n_] := a f[n - 1] (1 - f[n - 1]) corresponds to an iterated map of the kind discussed on page 920 , and has complicated behavior for many rational x .