As in one dimension, knowing the constraint does not immediately provide a procedure for finding a pattern which satisfies it.

Multiway systems [emulation] It is straightforward to emulate a k -color multiway system with a 2-color one, just by encoding successive colors by strings like "AAABBB" , "AAABAB" and "AABABB" that have no overlaps.

These rules only ever depend on one cell in each neighborhood.

Imagine a stadium full of people, with each person having two cards: one black and one white.

Such interfaces work well if what one wants is basically to take a single object and apply operations to it. … But whenever there is neither just a single active data element nor an obvious sequence of independent execution steps—as for many of the programs in this book—my experience has always been that the only viable choice of interface is a computer language like Mathematica, based essentially on one-dimensional sequences of word-like constructs.

Completeness [of number representations] If one successively reads 0's and 1's from an infinite sequence then the representations (c), (d) and (e) have the property that eventually one will always accumulate a valid representation for some number or another.

One explanation advanced by Albert Einstein was that the only physical laws we can recognize are ones that are easy to express in our system of mathematics.

Scope of existing sciences One might imagine that physics would for example concern itself with all aspects of physical systems, biology with all aspects of biological systems, and so on. … And when the phenomenon involves substantial complexity, what has in the past usually happened is that simpler and simpler aspects are investigated until one is found that is simple enough to analyze using the chosen way of thinking.

In doing mathematics there is often the idea that proofs should explain the result they prove—and one might not think this could be achieved if one just presents an object with certain properties.

Inevitably there are several scales involved, and one can only expect continuum behavior if one looks at scales intermediate between individual connections in the underlying network and the overall size of the whole network.