simple programs do than in all the previous ten years put together. … But with the overall framework I had developed I was gradually able to answer essentially all of what seemed to be the most obvious questions that I had raised.

And as described there, if such constraints can be satisfied at all, then it must be possible to satisfy them with a configuration that consists of a repetition of identical blocks. … In rule 90 (as well as other additive rules such as 60 and 150) any period can occur, but all configurations that repeat must consist of a sequence of identical blocks. … Rules 30 and 45 (as well as other one-sided additive rules) also have the property that all configurations that repeat must consist of a sequence of identical blocks.

But in essentially all cases they were viewed just as approximations to models based on traditional mathematical equations. … Nevertheless, starting in the 1970s many programs were written to simulate all sorts of scientific and technological systems, and often these programs in effect defined the models used. But in almost all cases the elements of the models were firmly based on traditional mathematical equations, and the programs themselves were highly complex, and not much like the simple programs I discuss in this book.

The results of this book indicate however that even programs that are very small—and thus have low algorithmic complexity—can nevertheless perform all sorts of complex computations.

If the coefficients inside all the sine functions are rational, then going from t = 0 to t = 2 π Apply[LCM, Map[Denominator, list]] yields a closed curve.

Each complete string in a multiway system corresponds to a possible slice that goes through all connections across a causal network.

The initial condition {1, 0, 1} with all cells 0 on the previous step yields a structure that repeats but only every 666 steps.

In all cases the cellular automaton rule, like the original operation on numbers, is invertible. … Non-trivial examples of multiplication by m in base k all appear to be class 3 systems (see page 250 ), with small changes in initial conditions growing at a roughly fixed rate.

Then in 1847 Ernst Kummer used ideas of factoring with algebraic integers to prove it for all n < 37 . Extensions of this method gradually allowed more cases to be covered, and by the 1990s computers had effectively given proofs for all n up to several million.

If too little is specified, there may be many possible solutions, while if too much is specified there may be no consistent solution at all. … In other PDEs, such as so-called elliptic ones, there is no such limit on the rate of information propagation, and as a result, it is immediately necessary to know values of u[t, x] at all x , and on the boundaries of the region, in order to determine u[t, x] for any t > 0 .