Implementation [of hexagonal cellular automata] One can treat hexagonal lattices as distorted square lattices, updated according to CAStep[rule_List, a_] := Map[rule 〚 14 - # 〛 &, a + 2 ListConvolve[{{1, 1, 0}, {1, 0, 1}, {0, 1, 1}}, a, 2], {2}] where rule = IntegerDigits[code, 2, 14] . … The centers of an array of regular hexagons are given by Table[{i √ 3 , j}, {i, 1, m}, {j, Mod[i, 2], n, 2}] .

between class 2 and class 3 in terms of what one might think of as overall activity. … And it turns out that class 4 systems with the same general characteristics are seen for example not only in ordinary cellular automata but also in such systems as continuous cellular automata. … The underlying rules in such systems involve a parameter that can vary smoothly from 0 to 1.

Statements in Peano arithmetic Examples include: • √ 2 is irrational: ¬ ∃ a ( ∃ b (b ≠ 0 ∧ a × a  ( Δ Δ 0) × (b × b))) • There are infinitely many primes of the form n 2 + 1 : ¬ ∃ n ( ∀ c ( ∃ a ( ∃ b (n + c) × (n + c) + Δ 0  ( Δ Δ a) × ( Δ Δ b)))) • Every even number (greater than 2) is the sum of two primes (Goldbach's Conjecture; see page 135 ): ∀ a ( ∃ b ( ∃ c (( Δ Δ 0) × ( Δ Δ a)  b + c ∧ ∀ d ( ∀ e ( ∀ f ((f  ( Δ Δ d) × ( Δ Δ e) ∨ f  Δ 0) ⇒ (f ≠ b ∧ f ≠ c))))))) The last two statements have never been proved true or false, and remain unsolved problems of number theory. The picture shows spacings between n for which n 2 + 1 is prime.

But in a practical computer, only a limited number of digits can ever be stored. … And so after some number of steps, all the digits in x are 0, and thus the value of x is simply 0. … (Base 10 is used so that multiplying for example 1/3 by 3 gives exactly 1 rather than the more confusing result 0.9999... obtained with base 2.)

In base 6, (3/2) n is a cellular automaton with rule {a_, b_, c_}  3 Mod[a + Quotient[b, 2], 2] + Quotient[3 Mod[b, 2] + Quotient[c, 2], 2] (Note that this rule is invertible.) … It is then possible to find special values of u (an example is 0.166669170371...) which make the first digit in the fractional part of u (3/2) n always nonzero, so that Mod[u (3/2) n , 1] > 1/6 . In general, it seems that Mod[u (3/2) n , 1] can be kept as large as about 0.30 (e.g. with u = 0.38906669065... ) but no larger.

The pictures below show paths obtained with the rule {1  {1}, 0  {0, 0, 1}} , starting from {0} . Note the similarity to the 2D system shown on page 190 . … But in a case like the rule {1  {0, 0, 1}, 0  {1, 0}} starting with {1} , the presence of many crossings tends to hide such regularity, as in the pictures below.

Tiling by polyominoes has been investigated since at least the late 1950s, particularly by Solomon Golomb , but it is only very recently that sets of polyominoes which force non-periodic patterns have been found. … At stage n the number of polyominoes of each type is Fibonacci[2n - {2, 0, 1}]/{1, 2, 1} .

Most of the programs require only the language component of Mathematica—and not its mathematical knowledge base—and so should run in all software systems powered by Mathematica, in which language capabilities are enabled. Here are examples of how some of the basic Mathematica constructs used in the notes in this book work: • Iteration Nest[f, x, 3] ⟶ f[f[f[x]]] NestList[f, x, 3] ⟶ {x, f[x], f[f[x]], f[f[f[x]]]} Fold[f, x, {1, 2}] ⟶ f[f[x, 1], 2] FoldList[f, x, {1, 2}] ⟶ {x, f[x, 1], f[f[x, 1], 2]} • Functional operations Function[x, x + k][a] ⟶ a + k (# + k&)[a] ⟶ a + k (r[#1] + s[#2]&)[a, b] ⟶ r[a] + s[b] Map[f, {a, b, c}] ⟶ {f[a], f[b], f[c]} Apply[f, {a, b, c}] ⟶ f[a, b, c] Select[{1, 2, 3, 4, 5}, EvenQ] ⟶ {2, 4} MapIndexed[f, {a, b, c}] ⟶ {f[a, {1}], f[b, {2}], f[c, {3}]} • List manipulation {a, b, c, d} 〚 3 〛 ⟶ c {a, b, c, d} 〚 {2, 4, 3, 2} 〛 ⟶ {b, d, c, b} Take[{a, b, c, d, e}, 2] ⟶ {a, b} Drop[{a, b, c, d, e}, -2] ⟶ {a, b, c} Rest[{a, b, c, d}] ⟶ {b, c, d} ReplacePart[{a, b, c, d}, x, 3] ⟶ {a, b, x, d} Length[{a, b, c}] ⟶ 3 Range[5] ⟶ {1, 2, 3, 4, 5} Table[f[i], {i, 4}] ⟶ {f[1], f[2], f[3], f[4]} Table[f[i, j], {i, 2}, {j, 3}] ⟶ {{f[1, 1], f[1, 2], f[1, 3]}, {f[2, 1], f[2, 2], f[2, 3]}} Array[f, {2, 2}] ⟶ {{f[1, 1], f[1, 2]}, {f[2, 1], f[2, 2]}} Flatten[{{a, b}, {c}, {d, e}}] ⟶ {a, b, c, d, e} Flatten[{{a, {b, c}}, {{d}, e}}, 1] ⟶ {a, {b, c}, {d}, e} Partition[{a, b, c, d}, 2, 1] ⟶ {{a, b}, {b, c}, {c, d}} Split[{a, a, a, b, b, a, a}] ⟶ {{a, a, a}, {b, b}, {a, a}} ListConvolve[{a, b}, {1, 2, 3, 4, 5}] ⟶ {2a + b, 3a + 2b, 4a + 3b, 5a + 4b} Position[{a, b, c, a, a}, a] ⟶ {{1}, {4}, {5}} RotateLeft[{a, b, c, d, e}, 2] ⟶ {c, d, e, a, b} Join[{a, b, c}, {d, b}] ⟶ {a, b, c, d, b} Union[{a, a, c, b, b}] ⟶ {a, b, c} • Transformation rules {a, b, c, d} /. b  p ⟶ {a, p, c, d} {f[a], f[b], f[c]} /. f[a]  p ⟶ {p, f[b], f[c]} {f[a], f[b], f[c]} /. f[x_]  p[x] ⟶ {p[a], p[b], p[c]} {f[1], f[b], f[2]} /. f[x_Integer]  p[x] ⟶ {p[1], f[b], p[2]} {f[1, 2], f[3], f[4, 5]} /. f[x_, y_]  x + y ⟶ {3, f[3], 9} {f[1], g[2], f[2], g[3]} /. f[1] | g[_]  p ⟶ {p, p, f[2], p} • Numerical functions Quotient[207, 10] ⟶ 20 Mod[207, 10] ⟶ 7 Floor[1.45] ⟶ 1 Ceiling[1.45] ⟶ 2 IntegerDigits[13, 2] ⟶ {1, 1, 0, 1} IntegerDigits[13, 2, 6] ⟶ {0, 0, 1, 1, 0, 1} DigitCount[13, 2, 1] ⟶ 3 FromDigits[{1, 1, 0, 1}, 2] ⟶ 13 The Mathematica programs in these notes are formatted in Mathematica StandardForm . The following table specifies how to enter these programs in Mathematica InputForm , using only ordinary keyboard characters:

Probably the simplest is a statement shown to be unprovable in Peano arithmetic by Laurence Kirby and Jeff Paris in 1982: that certain sequences g[n] defined by Reuben Goodstein in 1944 are of limited length for all n , where g[n_] := Map[First, NestWhileList[ {f[#] - 1, Last[#] + 1} &, {n, 3}, First[#] > 0 &]] f[{0, _}] = 0; f[{n_, k_}] := Apply[Plus, MapIndexed[#1 k^f[{#2 〚 1 〛 - 1, k}] &, Reverse[IntegerDigits[n, k - 1]]]] As in the pictures below, g[1] is {1, 0} , g[2] is {2, 2, 1, 0} and g[3] is {3, 3, 3, 2, 1, 0} . g[4] increases quadratically for a long time, with only element 3 × 2 402653211 - 2 finally being 0. … To cover all possible proofs, however, requires going up to the ordinal ε 0 . … But while it is known that in Peano arithmetic κ = ε 0 , quite how to describe the value of κ for, say, set theory remains unknown.

So to force particular forms of operators would require setting up axioms satisfied only by specific commutative groups. … (The groups can be written as products of cyclic ones whose orders correspond to the possible factors of n .)