(Other kinds of systematic searches done include ones for metal alloys, cooking recipes and plant hybrids.)

If one associates with each possible sequence of length n a number Sum[a i 2 -i , {i, n}] , then the set of sequences that actually occur at a given step form a Cantor set (see note below ), whose Hausdorff dimension turns out to be exactly h .

Self-gravitating systems The observed existence of structures such as galaxies might lead one to think that any large number of objects subject to mutual gravitational attraction might not follow the Second Law and become randomized, but might instead always form orderly clumps.

One is the almost-straight 30-mile railroad causeway built in 1959 that divides halves of the Great Salt Lake in Utah where the water is colored blue and orange.

Signals can sometimes travel through the magnetosphere along magnetic field lines from one hemisphere to the other, yielding so-called whistlers with frequencies that fall off in a highly regular way with time.

And in fact I argue that simple programs constructed without known purposes are what one needs to study to find the kinds of complex behavior we see.

And using Cantor's diagonal argument (see note below ) one can then show that this must be larger than ℵ 0 .

A triangular lattice is one example where they do not.

Especially after obtaining larger pictures in 1967, Ulam noted that in at least one case fairly simple growth rules generated a complicated pattern, and mentioned that this might be relevant to biology. … Despite the lack of investigation in science, one example of a cellular automaton did enter recreational computing in a major way in the early 1970s. … General 1D cellular automata are related to nonlinear feedback shift registers, and some explorations of these—including ones surprisingly close to rule 30 (see page 1088 )—were made usin

Universal Diophantine equation The equation is built up from ones whose solutions are set up to be integers that satisfy particular relations. … (Note that any variable a can be forced to be non-negative by including an equation a  w 2 + x 2 + y 2 + z 2 , as on page 910 .) … One can reduce such an exponential equation to a pure polynomial equation by encoding powers using integer equations.