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 ).

For the points that I make are often sufficiently complex to require quite long explanations. … In the main text I normally follow the principle that any paragraph should communicate just one basic idea. And my hope is then that after reading each paragraph readers will pause a moment to absorb each idea before going on to the next one.

Packing deformable objects If one pushes together identical deformable objects in 2D they tend to arrange themselves in a regular hexagonal array—and this configuration is known to minimize total boundary length. In 3D the arrangement one gets is typically not very regular—although as noted at various times since the 1600s individual objects often have pentagonal faces suggestive of dodecahedra. … Note that for a 3D Voronoi diagram with randomly placed points, the average number of faces for each region is 2 + 48 π 2 /35 ≃ 15.5 .)

And in the first case on the facing page , it so happens that the sequence of digits for each of the initial points shown is indeed quite random, so the behavior we see is correspondingly random. … And ultimately this question can only be answered by going outside of the system one is looking at, and studying whatever it was that set up its initial conditions.

One common mechanism is for a wave of a definite wavelength to form (see page 988 ), and then for some feature of each cycle of this wave to be picked out, as in the picture below. In Chladni figures of sand on vibrating plates and in cloud streets in the atmosphere what happens is that material collects at points of zero displacement.

Lagrange points and resonances often lead to simple geometrical patterns of orbiting bodies.

For purely algebraic questions of the kind that might arise in high-school algebra, however, one can use just the axioms given here. … With these axioms one can prove results about real polynomials, but not about arbitrary mathematical functions, or integers. … This is now in practice done by Simplify and other functions in Mathematica using methods of cylindrical algebraic decomposition invented in the 1970s—which work roughly by finding a succession of points of change using Resultant .

Every subsequent chapter in one way or another builds on earlier ones. … Note that in the main text I have tried to emphasize important points by various kinds of stylistic devices. … And in general these notes have a high enough information density that it will be rare that everything they say can readily be assimilated in just one reading, even if it is quite careful.

[Models involving] non-local processes It follows from the fact that any path in a finite network must always eventually return to a node where it has been before that any Markov process must be fundamentally local, in the sense that the probabilities it implies for what happens at a given point in a sequence must be independent of those for points sufficiently far away.

Even on this one page there are perhaps a dozen other very similar nested structures. 12 th century (Italian). … The pattern can be made by starting with a grid of triangles, then consistently pushing in or out the sides of each one. … One case with wood is Chinese lattice.