For one knows that given a single fixed underlying language, it is possible to describe an almost arbitrarily wide range of things. … The basic point is that if a system is universal, then it must effectively be capable of emulating any other system, and as a result it must be able to produce behavior that is as complex as the behavior of any other system.

But with the initial condition n = 512 , no repetition occurs for at least a million steps, at which point n has 568418 base 2 digits. … If one works directly with a digit sequence of fixed length, dropping any carries on the left, then a repetitive pattern is typically obtained fairly quickly.

But just how is it determined what section of the underlying genetic program should be used at what point in the development of the animal? … The existence of a fixed length scale at which such processes occur then almost inevitably implies that an embryo must develop in a somewhat hierarchical fashion.

The rules for such register machines are then idealizations of practical programs, and are taken to consist of fixed sequences of instructions, to be executed in turn. … But then, instead of just going on to execute the next instruction in the program, they jump to some specified other point in the program, and begin executing again from there.

But after just a few steps, the systems organize themselves to the point where definite structures become visible. … This particular structure is fairly simple: it just remains fixed in position and repeats every two steps.

However, the straightforward method for converting a t -digit number x to base k takes about t divisions, though this can be reduced to around Log[t] by using a recursive method such as FixedPoint[Flatten[Map[If[# < k, #, With[ {e = Ceiling[Log[k, #]/2]}, {Quotient[#, k e ], With[ {s = Mod[#, k e ]}, If[s  0, Table[0, {e}], {Table[0, {e - Floor[Log[k, s]] - 1}], s}]]}]] &, #]] &, {x}] The pictures below show stages in the computation of 3 20 (a) by a power tree in base 2 and (b) by conversion from base 3.

The address of the instruction to be fetched at each point is specified by the current value of the program counter—a number stored in memory that is incremented by the processor, or can be modified by instruction in the program. … Devices like keyboards, mice and microphones convert input into data that is inserted into memory at certain fixed locations. … A language provides a fixed set of constructs that allow one to specify computations.

The pattern corresponding to each point is the limit of Nest[Flatten[1 + {c #, Conjugate[c] #}]&, {1}, n] when n  ∞ . … Every point in the pattern must correspond to some list of left and right branchings, represented by 0's and 1's respectively; in terms of this list the position of the point is given by Fold[1 + {c, Conjugate[c]} 〚 1 + #2 〛 #1&, 1, Reverse[list]] . … A simple way to approximate the pictures in the main text would be to generate patterns by iterating the substitution system a fixed number of times.

The same convergence to a single fixed point is observed. … (To recover correct infinite size results one must increase size while keeping the number of steps of evolution fixed; the networks shown above, however, effectively depend on arbitrarily many steps of evolution.)

But the crucial point is that this will not take long to happen throughout any network if it is appropriately connected. Traditional models tend to assume that there are ultimately a fixed number of spacetime dimensions in the universe. … 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.