Successive steps in the iterative procedure used on this page are given by Move[list_] := (If[Cost[#] < Cost[list], #, list] &)[ MapAt[1 - # &, list, Random[Integer, {1, Length[list]}]]] while those in the procedure on page 347 have ≤ in place of < .

Note that tree-like snowflakes are what make snow fluffy, while simple hexagons make it denser and more slippery.

Discrete quantum mechanics While there are many issues in finding a complete underlying discrete model for quantum phenomena, it is quite straightforward to set up continuous cellular automata whose limiting behavior reproduces the evolution of probability amplitudes in standard quantum mechanics. … Despite their basic setup the systems discussed here are not direct analogs of standard quantum spin systems, since these normally have local Hamiltonians and non-local evolution functions, while the systems here have local evolution functions but seem always to require non-local Hamiltonians.)

The system here can be represented by the rule n  If[EvenQ[n], 3n/2, 3(n + 1)/2] , while the one on page 100 follows the rule n  If[EvenQ[n], 3n/2, (3n + 1)/2] .

Continuous Versus Discrete Systems One of the most obvious differences between my approach to science based on simple programs and the traditional approach based on mathematical equations is that programs tend to involve discrete elements while equations tend to involve continuous quantities.

The simplest non-trivial pair of blocks that has this property is , , while the simplest triple is , , .

But while this means that it might be possible for there to be arbitrariness in the causal network for the universe, it still tends to be my suspicion that there is not—and that in fact the particular rules followed by the universe do in the end have the property that they always yield the same causal network.

Network (a) specifies that the colors of successive squares should be chosen independently, while network (b) specifies that this should be done for successive pairs of squares.

One might imagine perhaps that while there could in principle be methods of perception that would recognize features beyond, say, repetition and nesting, any single such feature might never occur in a sufficiently wide range of systems to make its recognition generally useful to a biological organism.

But while this is already a remarkable result, it represents only a first step in the direction of the Principle of Computational