In the pictures below, the n th point has position ( √ n {Sin[#], Cos[#]} &)[2 π n GoldenRatio] , and in such pictures regular spirals or parastichies emanating from the center are seen whenever points whose numbers differ by Fibonacci[m] are joined.

The basic origin of this phenomenon is the averaging effect of randomness discussed in Chapter 7 (technically, it is the survival only of leading operators at renormalization group fixed points).

that after going through perhaps nine trillion Turing machines one will definitely tend to find an example that is universal. But presumably one will actually find examples much earlier—since for example the 2-state 3-color machine on page 709 is only number 596,440. … With short initial conditions, the pictures at the top of the next page demonstrate that combinators tend to evolve quickly to simple fixed points.

Most important among these is that any material one works with will presumably be made of atoms. … One might think that atoms would always be so small that their size would in practice be irrelevant. … And indeed after just thirty steps, the description of the kneading process given above would imply that two points initially only one atom apart would end up nearly a meter apart.

For odd n up to 500 million the only values near 0 that appear in the curve are {-6, -5, -4, -2, -1, 6, 18, 26, 30, 36} , with, for example, the first 6 occurring at n = 8925 and last 18 occurring at n = 159030135 .

But in the abstract there is no reason that these connections should lead to points that can in any way be viewed as nearby in space. … One might choose to consider systems like these just because it seems easier to specify their rules. But their locality also seems important in giving rise to anything that one can reasonably recognize as space.

Nucleation [of crystals] In the absence of container walls or of other objects that can act as seeds, liquids and gases can typically be supercooled quite far below their freezing points.

The result of this is that points in space can always be specified by lists of coordinates—although historically one of the objectives of differential geometry has been to find ways to define properties like curvature so that they do not depend on the choice of such coordinates. … One is based on looking at infinitesimal vectors u , v and w and asking how much w differs when transported two ways around the edges of a parallelogram, from x to x + u + v via x + u and via x + v . In ordinary flat space there is no difference, but in general the difference is a vector that is defined to be Riemann . u . v. w .

My explanation of the Second Law What I say in this book is not incompatible with much of what has been said about the Second Law before; it is simply that I make more definite some key points that have been left vague before.

For example, to find a definite volume growth rate one does still need to take some kind of limit—and one needs to avoid sampling too many or too few nodes in the network. … (Note that given explicit coordinates, one can check whether one is in d or more dimensions by asking for all possible points Det[Table[(x[i] - x[j]) . (x[i] - x[j]), {i, d + 3}, {j, d + 3}]]  0 and this should also work for sufficiently separated points on networks.