The top row shows how a particular non-deterministic Turing machine behaves with successive sequences of choices for rules to apply.

Ornamental art Almost all major cultural periods are associated with certain characteristic forms of ornament. … The pattern—in either its square or rounded form—has appeared with remarkably little variation in a huge variety of places all over the world—from Cretan coins, to graffiti at Pompeii, to the floor of the cathedral at Chartres, to carvings in Peru, to logos for aboriginal tribes. … The patterns shown are all basically two-dimensional.

The top two (both with 120 comparisons) have a repetitive structure and correspond to standard sorting algorithms: transposition sort and insertion sort. … (In general all lists will be sorted correctly if lists of just 0's and 1's are sorted correctly; allowing even just one of these 2 n cases to be wrong greatly reduces the number of comparisons needed.)

The pictures below show results for a fairly typical sequence of initial conditions where all three bodies interact. (The two bodies at the bottom are initially at rest; the body at the top is given progressively larger rightward velocities.) … Often this happens quickly, but sometimes all three bodies show complex and apparently random behavior for quite a while.

Six types are known: u , d , c (charm), s (strange), t (top), b . … Apart from the photon (and graviton), all have distinct antiparticles. … Grand unified models typically do this for all known gauge bosons (except gravitons) and for corresponding families of quarks and leptons—and inevitably imply the existence of various additional particles more massive than those known, but with properties that are somehow intermediate.

We would not usually say, therefore, that either of the first two pictures at the top of the facing page seem random, since we can readily recognize highly regular repetitive and nested patterns in them.

The picture at the top of the facing page shows what happens if one considers two steps of cellular automaton evolution.

Note that if the elements of a surface are allowed to change shape, then the surface can always remain flat, as in the top row of pictures on page 412 .

Larger objects normally come to the top (as with mixed nuts, popcorn or pebbles and sand), essentially because the smaller ones more easily fall through interstices.

But there is a fundamental result in graph theory that shows that if a network is not planar, then it must always be possible to identify in it a specific part that can be reduced to one of the two forms shown in the top picture—or just the second form for a network with three connections at each node.