Note that to generate the pictures that follow requires applying the underlying cellular automaton rule for individual cells a total of about 12 million times.

Note that to generate the pictures that follow requires applying the underlying cellular automaton rule for individual cells a total of about 12 million times.

Semigroups [and axioms] Despite their simpler definition, semigroups have been much less studied than groups, and there have for example been about 7 times fewer mathematical publications about them (and another 7 times fewer about monoids).

Instead, the possibility of motion that leads to earlier times simply implies a requirement of consistency between behavior at earlier and later times.

The total number of commutative groups with k elements is just Apply[Times, Map[PartitionsP[Last[#]] &, FactorInteger[k]]] (Relabelling of elements makes the number of possible operator forms up to k! times larger.)

However, as was shown in the 1800s, this method can yield only numbers formed by operating on rationals with combinations of Plus , Times and Sqrt . (Thus it is impossible with ruler and compass to construct π and "square the circle" but it is possible to construct 17-gons or other n -gons for which FunctionExpand[Sin[ π /n]] contains only Plus , Times and Sqrt .) … Note that above degree 4, algebraic numbers cannot in general be expressed in radicals involving only Plus , Times and Power (see page 945 ).

f[n_] := n f[n - 1]; f[1] = 1 f[n_] := Product[i, {i, n}] f[n_] := Module[{t = 1}, Do[t = t i, {i, n}]; t] f[n_] := Module[{t = 1, i}, For[i = 1, i ≤ n, i++, t ⋆ = i]; t] f[n_] := Apply[Times, Range[n]] f[n_] := Fold[Times, 1, Range[n]] f[n_] := If[n  1, 1, n f[n - 1]] f[n_] := Fold[#2[#1] &, 1, Array[Function[t, # t] &, n]] f = If[#1  1, 1, #1 #0[#1 - 1]] &

And in more recent times sensitivity to initial conditions and quantum randomness have been proposed as more appropriate scientific explanations. … For as we have seen many times in this book even systems with quite simple and definite underlying rules can produce behavior so complex that it seems free of obvious rules.

It is mentioned several times in the Bible, and even today remains the most common method for large lotteries. … Children's games like musical chairs in effect generate randomness by picking arbitrary stopping times. Games of chance based on wheels seem to have existed in Roman times; roulette developed in the 1700s.

But perhaps most crucial for me was that the process was a bit like what I have ended up doing countless times in designing Mathematica: start from elaborate technical ideas, then gradually see how to capture their essential features in something amazingly simple. And the fact that I had managed to make this work so many times in Mathematica was part of what gave me the confidence to try doing something similar in all sorts of areas of science.