(K 5 and K 3,3 are examples of so-called complete graphs, obtained by taking sets of specified numbers of nodes and connecting them in all possible ways.) … (There is in fact a general theorem established since the 1980s that absolutely any list of networks—say for example ones that cannot be laid on a given surface—must actually in effect always all be reducible to some finite list of minors.)

In terms of multiway systems, each of the elements corresponds to a disconnected part of the network formed from all possible sequences. … The so-called free semigroup has no relations and thus no rules, so that all strings of generators correspond to distinct elements, and the Cayley graph is a tree like the ones shown on page 196 . … A major mathematical achievement in the 1980s was the complete classification of all possible so-called simple finite groups that in effect have no factors.

One can characterize the symmetry of a pattern by taking the list v of positions of cells it contains, and looking at tensors of successive ranks n : Apply[Plus, Map[Apply[Outer[Times, ##] &, Table[#, {n}]] &, v]] For circular or spherical patterns that are perfectly isotropic in d dimensions these tensors must all be proportional to (d - 2)!!… On a 2D lattice with m directions, all moments are forced to be zero except when m divides n . … And at least in the case of ordinary random walks, they do, so that for example, the ratio averaged over all possible walks of n = 4 tensor components after t steps on a square lattice is β = 3 + 2/(t - 1) , converging to the isotropic value 3, and the ratio of n = 6 components is 5 - 4/(t - 1) + 32/(3t - 4) .

And to capture all these eddies in a computation eventually involves prohibitively large amounts of information. … The Navier–Stokes equations assume that all speeds are small compared to the speed of sound—and thus that the Mach number giving the ratio of these speeds is much less than one. In essentially all practical situations, Mach numbers close to one occur only at extremely high Reynolds numbers—where turbulence in any case would make it impossible to work out the detailed consequences of the Navier–Stokes equations.

And in terms of Feynman diagrams the way this happens is by virtual particle-antiparticle pairs of all types and all energy-momenta continually forming and annihilating at all points in the vacuum.

Leading digits [in numbers] Even though in individual numbers generated by simple mathematical procedures all possible digits often appear to occur with equal frequency, leading digits in sequences of numbers typically do not.

Three sine functions All zeros of the function Sin[a x] + Sin[b x] lie on the real axis.

Non-overlapping [network] clusters The picture shows all distinct clusters with 3 dangling connections and 9 nodes that are not self-overlapping.

Other reversible systems Reversible examples can be found of essentially all the types of systems discussed in this book.

But to apply such an argument one must among other things assume that we can imagine all the ways in which intelligence could conceivably operate.