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).

Parameter space sets Points in the space of parameters can conveniently be labelled by a complex number c , where the imaginary direction is taken to increase to the right. … In practice, however, it is essential to prune the tree of points at each stage. And at least for Abs[c] not too close to 1, this can be done by discarding points that are so far away from the peephole that their descendents could not possibly return to it.

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.

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.

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 ).

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 ).

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.

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 .

Long halting times [in symbolic systems] Symbolic systems with rules of the form ℯ [x_][y_]  Nest[x, y, r] always evolve to fixed points—though with initial conditions of size n this can take of order Nest[r # &, 0, n] steps (see above ).