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The normalization of probabilities then implies p = 1/(2k) , and since the word at rank roughly k m then has probability 1/(2k) m , Zipf's law follows.
For it implies that there is never an immediate reason to go beyond studying systems with rather simple underlying rules.
For in addition to opening up vast new domains of exploration, it implies a radical rethinking of how processes in nature and elsewhere work.
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.
And what we see immediately from these pictures is that while some systems exhibit exactly the kind of randomization implied by the Second Law, others do not.
But if one identifies time with position down the page, the presence of connections that go up as well as down the page implies that in some sense time does not always progress in the same direction.
So this implies that one can in fact meaningfully associate a definite structure with non-planarity.
For my model of particles such as electrons being persistent structures in a network might initially seem to imply that such particles are somehow definite objects just like ones familiar from everyday experience.
In general, the equations of quantum field theory seem to imply that there can be all sorts of complicated configurations in the field, even in the absence of actual particles.
For among other things, it implies that the universal cellular automaton can emulate cellular automata whose rules are more complicated than its own.