And what one finds is that in certain cases—notably in connection with nesting at critical points associated with phase transitions (see page 981 )—certain averages turn out to be the same as one would get if one did no blocking but just changed parameters ("coupling constants") in the underlying rules that specify the weighting of different configurations. … And when one looks at large scales the versions of these equations that arise in practice essentially always show fixed points, whose properties do not depend much on details of the equations—leading to certain universal results across many different underlying systems (see page 983 ). … And in most cases such rules will not suffice even if one takes averages.

A typical kind of failure, illustrated in the pictures on the next page , is that points with coordinates determined by successive numbers from the generator turn out to be distributed in an embarrassingly regular way. At first, such failures might suggest that more complicated schemes must be needed if one is to get good randomness. … And seeing this one might conclude that it must be essentially impossible to produce good randomness with any kind of system that has reasonably simple rules.

To set up general relativity one then considers not Riemannian manifolds but instead Lorentzian ones in which the metric is not positive definite, but instead has the signature of Minkowski space. … For given any point, even the light cone that corresponds to points at zero spacetime distance from it has an infinite volume. … Then one defines a cone of height t whose apex is a given point to be those points whose displacement vector v satisfies 0 > e . g . v > -t (and 0 > v . g. v ).

Yet these paths can still be the shortest—or so-called geodesics—if one takes space to be curved. And indeed if space has appropriate curvature one can get all sorts of paths, as in the pictures below. … The paths are geodesics which go the minimum distance on the surface to get to all the points they reach.

Given the network for a particular n , it is straightforward to see what happens when only certain length n blocks are allowed: one just keeps the arcs in the network that correspond to allowed blocks, and drops all other ones. Then if one can still form an infinite path by going along the arcs that remain, this path will correspond to a pattern that satisfies the constraints. … But the crucial point is that since there are only k n - 1 nodes in the network, then if any infinite path is possible, there must be such a path that visits the same node and thus repeats itself after at most k n - 1 cells.

For strings the analogous problem is straightforward, since in a string of length n one can ultimately just try each of the n possible starting points for the substring and see for which of them a match occurs. But for a network with n nodes, a similar procedure would require one to check n k possible configurations in order to find out where a subnetwork of size k occurs. In practice, however, for fixed subnetworks, one can devise fairly efficient procedures.

Standard treatment [of relativity] In a standard treatment of relativity theory one way to begin is to consider setting up a square grid of points in space and time—and then to ask what kind of transformed grid corresponds to this same set of points if one is moving at some velocity v . At first one might assume that the answer would just be a grid that has been sheared by the simple transformation {t, x}  {t, x - v t} , as in the first row of pictures below.

General topology [and axioms] The axioms given define properties of open sets of points in spaces—and in effect allow issues like connectivity and continuity to be discussed in terms of set theory without introducing any explicit distance function.

And as a result, one might think that digits which lie far to the right in the initial conditions would never be important. … Indeed in many ways the only real difference is that instead of The digit sequences of positions of points on successive steps in the two examples of kneading processes at the bottom of the previous page . At each step these digit sequences are shifted one place to the left.

In each case the result of 10 steps of evolution is shown, and the pictures are scaled so that all points above the bottom of the original stem can be included.