Intrinsically defined curves With curvature given by a function f[s] of the arc length s , explicit coordinates {x[s], y[s]} of points are obtained from (compare page 1048 ) NDSolve[{x'[s]  Cos[ θ [s]], y'[s]  Sin[ θ [s]], θ '[s]  f[s], x[0]  y[0]  θ [0]  0}, {x, y, θ }, {s, 0, s max }] For various choices of f[s] , formulas for {x[s], y[s]} can be found using DSolve : f[s] = 1: {Sin[ θ ], Cos[ θ ]} f[s] = s: {FresnelS[ θ ], FresnelC[ θ ]} f[s] = 1/ √ s : √ θ {Sin[ √ θ ], Cos[ √ θ ]} f[s] = 1/s: θ {Cos[Log[ θ ]], Sin[Log[ θ ]]} f[s] = 1/s 2 : θ {Sin[1/ θ ], Cos[1/ θ ]} f[s] = s n : result involves Gamma[1/n, ±  θ n/n ] f[s] = Sin[s] : result involves Integrate[Sin[Sin[ θ ]], θ ] , expressible in terms of generalized Kampé de Fériet hypergeometric functions of two variables. When s max  ∞ , f[s] = a s Sin[s] yields 2D shapes that are basically nested, with pieces overlapping for Abs[a] < 1 .

A way to find candidates for q is to compute NullSpace[Table[With[{u = Table[Random[Integer, {0, k - 1}], {m}]}, BC[CAStep[u]] - BC[u]], {s}]] for progressively larger m and s , and to see what lists continue to appear. … (With k = 2 , for b = 1 , {1, 1} represents conservation of the total number of cells, regardless of color, while for b = 2 , {1, 1, 1, 1} represents the same thing, while {0, 1, -1, 0} represents the fact that in going along in any state the number of black-to-white transitions must equal the number of white-to-black ones.) … To identify any such quantity with certainty, it turns out to be enough to look at the k b + 2r - 1 states where no block of length b + 2r - 1 appears more than once (and perhaps even just some fairly small subset of these).

Such a circle has area 2 π a 2 (1 - Cos[r/a]) = π r 2 (1 - r 2 /(12 a 2 ) + r 4 /(360a 4 ) - …) In the d -dimensional space corresponding to the surface of a (d + 1) -dimensional sphere of radius a , the volume of a d -dimensional sphere of radius r is similarly given by d s[d] a d Integrate[Sin[ θ ] d - 1 , { θ ,0, r/a}] = s[d] r d (1 - d (d - 1) r 2 /((6 (d + 2))a 2 + (d (5d 2 - 12d + 7))r 4 /((360 (d + 4))a 4 ) …) where Integrate[Sin[x] d - 1 , x] = -Cos[x] Hypergeometric2F1[1/2, (2 - d)/2, 3/2, Cos[x] 2 ] In an arbitrary d -dimensional space the volume of a sphere can depend on position, but in general it is given by s[d] r d (1 - RicciScalar r 2 /(6(d + 2)) + …) where the Ricci scalar curvature is evaluated at the position of the sphere. … But in normal coordinates the first non-trivial term in the expansion of the metric is proportional to the Riemann tensor, yet the symmetry of a spherical volume makes it inevitable that the Ricci scalar is the only combination of components that can appear at lowest order. … Map[ ∂ # f &, p]), p]/Sqrt[Det[g]] In general the series in r may not converge, but it is known that at least in most cases only flat space can give a result that shows no correction to the basic r d form.

Mobile automata [from cellular automata] Given a mobile automaton with rules in the form used on page 887 , a cellular automaton which emulates it can be constructed using MAToCA[rules_] := Append[Flatten[Map[g, rules]], {_, _, x_, _, _}  x] g[{a_, b_, c_}  {d_, e_}] := {{_, a, b + 2, c, _}  d, If[e  1, {a, b + 2, c, _, _}  c + 2, {_, _, a, b + 2, c}  a + 2]} This specific definition assumes that the mobile automaton has two possible colors for each cell; it yields a cellular automaton with four possible colors for each cell. An initial condition with a single 2 surrounded by 0's corresponds to all cells being white in the mobile automaton.

For equations of the form ∂ tt u[t, x]  ∂ xx u[t, x] + f[u[t, x]] one can set up a simple finite difference method by taking f in the form of pure function and creating from it a kernel with space step dx and time step dt : PDEKernel[f_, {dx_, dt_}] := Compile[{a,b,c,d}, Evaluate[(2 b - d) + ((a + c - 2 b)/dx 2 + f[b]) dt 2 ]] Iteration for n steps is then performed by PDEEvolveList[ker_, {u0_, u1_}, n_] := Map[First, NestList[PDEStep[ker, #]&, {u0, u1}, n]] PDEStep[ker_, {u1_, u2_}] := {u2, Apply[ker, Transpose[ {RotateLeft[u2], u2, RotateRight[u2], u1}], {1}]} With this approach an approximation to the top example on page 165 can be obtained from PDEEvolveList[PDEKernel[ (1 - # 2 )(1 + #)&, {.1, .05}], Transpose[ Table[{1, 1} N[Exp[-x 2 ]], {x, -20, 20, .1}]], 400] For both this example and the middle one the results converge rapidly as dx decreases.

The base 2 digit sequences obtained with this map starting from x = 1/8 are shown below for various values of a . … The detailed behavior is different for every value of a , but whenever the repetition period is 2 j , it turns out that with any initial condition the leftmost digit always eventually follows a sequence that consists of repetitions of step j in the evolution of the substitution system {1  {1, 0}, 0  {1, 1}} starting either from {0} or {1} . … In the special case a = 4 , it turns out that replacing x by Sin[ π u] 2 makes the mapping become just u  FractionalPart[2 u] , revealing simple shift map dependence on the initial digit sequence.

Quadratic residue sequences As an outgrowth of ideas related to RSA cryptography it was shown in 1982 by Lenore Blum , Manuel Blum and Michael Shub that the sequence Mod[NestList[Mod[# 2 , m] &, x0, n], 2] discussed on page 975 has the property that if m=p q with p and q primes (congruent to 3 modulo 4) then any systematic regularities detected in the sequence can eventually be used to discover factors of m . What is behind this is that each of the numbers in the basic sequence here must be a so-called quadratic residue of the form Mod[v 2 , m] , and given any such quadratic residue x the expression GCD[x + Mod[x 2 , m], m] turns out always to be a factor of m —and at least sometimes a non-trivial one. … The condition Mod[p, 4]  Mod[q, 4]  3 ensures that only one of the solutions +v and -v to x  Mod[v 2 , m] is ever a quadratic residue, with the result that the iterated mapping x  Mod[x 2 , m] always has a unique inverse.

The main point, I believe, is that in both the systems it studies and the questions it asks mathematics is much more a product of its history than is usually realized.

Complex powers [of numbers] The pictures below show successive powers of complex numbers z with digits extracted according to (2 d[Re[#], w] + d[Im[#], w]) & [z t ] d[x_, w_] := If[x < 0, 1 - d[-x, w], IntegerDigits[x, 2, w]] Non-trivial cases of complex number multiplication never correspond to local cellular automaton operations.

Implementation [of operators from axioms] Given an axiom system in the form {f[a, f[a, a]]  a, f[a, b]  f[b, a]} one can find rule numbers for the operators f[x, y] with k values for each variable that are consistent with the axiom system by using Module[{c, v}, c = Apply[Function, {v = Union[Level[axioms, {-1}]], Apply[And, axioms]}]; Select[Range[0, k k 2 - 1], With[{u = IntegerDigits[#, k, k 2 ]}, Block[{f}, f[x_, y_] := u 〚 -1 - k x - y 〛 ; Array[c, Table[k, {Length[v]}], 0, And]]] &]] For k = 4 this involves checking nearly 16 4 or 4 billion cases, though many of these can often be avoided, for example by using analogs of the so-called Davis–Putnam rules.