But it turns out that there can still be other kinds of curvature—described for example by the so-called Riemann tensor—and these can in fact lead to all sorts of phenomena. … So this means that to know what will happen even in phenomena primarily associated with gravity one typically has to know all sorts of properties of matter. But why exactly does matter have to be introduced explicitly at all?

But the fact remains that despite all the various methods of mathematical and other analysis that have been developed, our visual system still represents one of the most powerful and reliable tools we have. And certainly in writing this book I have relied heavily on our ability to make all sorts of deductions on the basis of looking at visual representations. … And this is why, for example, we have the impression that mixtures of just three fixed colors can successfully reproduce all other colors.

And perhaps as a result of this, it has sometimes been thought that if one could just compute frequencies of blocks of all lengths one would have a kind of universal test for randomness. … For in each case all one has to do is to compute the value of a quantity from a particular sequence of data, and then compare this value with what would be obtained by averaging over all possible sequences. … Indeed, beyond block frequencies, the only other ones that are common are those based on correlations, spectra, and occasionally run lengths—all of which we already discussed earlier in this chapter .

And what one sees is that there is a broad distribution: from cases in which very few strings can be generated—corresponding to a very incomplete axiom system—to cases in which all or almost all strings can be generated—corresponding to a very inconsistent axiom system. … For if one looks at axiom systems that are widely used in mathematics they almost all tend to be complete enough to prove at least a fair fraction of statements either true or false. … I suspect that it has to do with the fact that in mathematics one usually wants axiom systems that one can think of as somehow describing definite kinds of objects—about which one then expects to be able to establish all sorts of definite statements.

there are all sorts of features in the behavior of these rules that could in principle represent a possible purpose. … So what does all this mean for extraterrestrial intelligence? … For certainly our astronomical observations have revealed all sorts of phenomena for which we do not yet have any very satisfactory explanations.

This means that every block with length up to n (except all 0's) must occur with equal frequency. (Note that only a small fraction of all possible sequences with this property can be generated by LFSRs.) … This quantity is -1 for all nonzero m for PN sequences (so that all but the first component in Abs[Fourier[(-1) list ]] 2 are equal), but has mean 0 for truly random sequences.

[No text on this page] The behavior of all cellular automata that involve only nearest neighbors in a symmetrical way, have two possible colors for each cell, and leave states consisting only of white cells unchanged.

Indeed, in all the more than two hundred billion digits of π that have so far been computed, no significant regularity of any kind has ever been found. … In all cases what we see is that the digit sequences of such numbers have a simple repetitive form. And in fact, it turns out that absolutely all rational numbers have digit sequences that eventually repeat.

If one looks at many multiway systems, most either grow exponentially quickly, or not at all; slow growth of the kind seen on the facing page is rather rare. … If one allows more rapid growth, however, then there presumably start to be all sorts of multiway systems that never show any such regularity. … The particular rule used here eventually generates all states beginning with a white cell.

And in almost all of these, there is no overlap not only within a single cluster, but also between different clusters. … One feature of the various rules I showed earlier is that they all maintain planarity of networks—so that if one starts with a network that can be laid out in the plane without any lines crossing, then every subsequent network one gets will also have this property. … But this very randomness will most likely be what for example allows a definite and robust value of 3 to emerge for the dimensionality of space—even though all of the many complicated phenomena in our universe must also somehow be represented within the structure of the same network.