But although it so happens that essentially the same mathematical notation is in practice used all around the world by speakers of every ordinary language, I do not believe that it is in any way unique or inevitable, and in fact I think it shows most of the same issues of dependence on history and context as any ordinary language. … The idea of having notation for operators emerged in the early 1600s, and by the end of the 1600s, notably with the work of Gottfried Leibniz , essentially all the basic notation now used in algebra and calculus had been established. … And while traditional mathematical notation suffers from some inconsistencies and ambiguities, it was possible in developing Mathematica StandardForm to set up something very close that can be interpreted uniquely in all cases.

Around 1921, Post apparently studied all tag systems of his type that involve removal and addition of no more than two elements at each step, and he concluded that none of them produced complicated behavior. … But at least for all the initial conditions up to length 28, the rule eventually just leads to behavior that repeats with a period of 1, 2, 6, 10, 28 or 40.

Smooth iterated maps In the main text, all the functions used as mappings consist of linear pieces, usually joined together discontinuously. … (An important result discovered by Mitchell Feigenbaum in 1975 is that this basic setup is universal to all smooth maps whose functions have a single hump.)

The correct answer is normally assumed to be the one that in a sense allows the simplest description of all the data.

Shown on a logarithmic scale, representations (b) through (e) (given here for numbers 1 through 500) all grow roughly linearly:

An initial condition with a single 2 surrounded by 0's corresponds to all cells being white in the mobile automaton.

And in practice if there are many simple rules that are universal they cannot all be related by simple encodings.

The results in this book establish a new point of view somewhere between all of these.

With all cells 0 on one step, and a block of nonzero cells on the next step, the periods are for example: {1} : 21 ; {1, 1} : 3n - 8 ; {1, 0, 1} : 666 ; {1, 1, 1} : 3n - 8 ; {1, 0, 0, 1} : irregular ( < 24n ; peaks at 6j + 1 ); {1, 0, 0, 1, 0, 1} : irregular ( ≲ 2 n ; 857727 for n= 2 6 ; 13705406 for n = 100 ).

Pictures in the book All the diagrammatic pictures in this book were created using Mathematica.