Leaf shapes Leaves are usually put into categories like the ones below, with names mostly derived from Latin words for similar-looking objects. … Traditional evolutionary explanations have not had much to say about detailed questions of leaf shape; one minor claim is that the pointed tips at the ends of many tropical leaves exist to allow moisture to drip off the leaves.

I searched the scientific literature, talked to many people, and found out that systems similar to the ones I was studying had been named "cellular automata" some thirty years earlier.

Many musical instruments produce sound by vibrating strings or air in cylindrical or conical tubes, and in these cases, there is one main frequency, together with roughly equally spaced overtones.

One approach extrapolates chip technology, and studies placing atoms individually on solid surfaces using for example scanning probe microscopy.

.) • P (polynomial time): can be solved (with one processor) in a number of steps that increases like a polynomial in the input size.

In fact, as mentioned on page 1155 , if one does not allow lemmas some proofs perhaps have to become exponentially longer.

In fact, much as for cellular automata, more explicit experiments have been done on 2D Turing machines than 1D ones.

And in general if there is a path from x to y then one writes x ≻ y .

And by early 1985 I had written what I consider to be my two most fundamental (if excessively short) papers from the period: one on undecidability and intractability in theoretical physics, and the other on intrinsic randomness generation and the origins of randomness in physical systems. … In June 1986 I organized one last conference on cellular automata—then in August 1986 essentially left the field to begin the development of Mathematica. … But cellular automata—and especially 1D ones—make the phenomena particularly clear, which is why even after investigating all sorts of other systems 1D cellular automata are still the most common examples that I use in this book.

Note that a given program can essentially always be written in Mathematica in many different ways—though often other ways end up being vastly longer than the ones presented here.