The pictures on the right are obtained by keeping only the steps indicated by arrows on the left, corresponding to times when the active cell in the mobile automaton is further to the left than it has ever been before.

Evaluation chains The idea of building up computations like 1 + 1 + 1 + … from partial results has existed since Egyptian times.

Longest halting times [in Turing machines] The pictures below show the largest numbers of steps t[x] that it takes any machine of a particular type to halt when given successive inputs x .

The many different possible external forms of crystals have been noted in mineralogy since Greek times, but although classification schemes have been given, these forms have apparently still not been studied in a particularly systematic way.

In the past century features of phyllotaxis have been rediscovered surprisingly many times, with work being done quite independently both in abstract mathematical settings, and in the context of specific models (most of which are ultimately very similar).

But in general it may be undecidable even whether two such forms are actually equivalent (compare the notes below and on page 1051 )—since to tell this one might need to be able to apply the rules infinitely many times.

Out of the many hundreds of times that I have used hashing in practice, I recall only a couple of cases where schemes like the one just described were not adequate, and in these cases the data always turned out to have quite dramatic regularities. … Hashing is a sufficiently simple idea that it has been invented independently many times since at least the 1950s.

The top plot is exactly for case (b); the bottom one is for a system three times larger in size.

For given the state of a system, they provide rules for determining its state at subsequent times.

At times the role of mathematics in science has been used in philosophy as an indicator of the ultimate power of human thinking.