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But the discovery in this book that a wide range of systems can generate randomness even with very simple initial conditions makes it seem considerably less surprising.
So long as f[n] grows less rapidly than 2 n (as when f = Fibonacci or f = Prime ), digits 0 and 1 will suffice, though the representation is not generally unique.
But in 1993 Denis Weaire and Robert Phelan discovered a layered repetitive arrangement of 12- and 14-faced polyhedra (average 13.5) that yields 0.003 times less total area. It seems likely that there are polyhedra which fill space in a less regular way and yield still smaller total area.
One cannot for example sort n objects in less than about n steps since one must at least look at each object, and one cannot multiply two n -digit numbers in less than about n steps since one must at least look at each digit. … It is fairly clear that this cannot be done in less than about s steps.
One might think that as such capabilities increase, data compression would become less relevant.
Observed phyllotaxis
Many spiral patterns in actual plants converge to within a degree or less of 137.5°, though just as in the model in the main text, there are usually deviations for the first few elements produced.
In 1890 Henri Poincaré established the somewhat less obvious fact that even continuous systems also always eventually get at least arbitrarily close to repeating themselves.
No persistent structures of any size exist in this cellular automaton with repetition periods of less than 5 steps.
But as their names suggest Simplify and FullSimplify were intended to be less predictable—and just to do what they can and then return a result.
Semigroups [and axioms]
Despite their simpler definition, semigroups have been much less studied than groups, and there have for example been about 7 times fewer mathematical publications about them (and another 7 times fewer about monoids).