Of regular polygons, only squares, triangles and hexagons can be used to do this, and in these cases the tilings are always repetitive. For some time it was believed that any set of tiles that could cover the plane could be arranged to do so repetitively.

But as a practical matter it does not. … But in fact the cosmic microwave background in effect does exactly this.

But the rules that win most often typically seem to do so in rather simple ways.

In general, materials with nested structures do not necessarily yield discrete diffraction patterns.

The sequence {1, 2, 2, 1, 1, 2, …} defined by the property list  Map[Length, Split[list]] was suggested as a mathematical puzzle by William Kolakoski in 1965 and is equivalent to Join[{1, 2}, Map[First, CTEvolveList[{{1}, {2}}, {2}, t]]] It is known that this sequence does not repeat, contains no more than two identical consecutive blocks, and has at least very close to equal numbers of 1's and 2's.

Nevertheless, the procedures discussed in this section do not manage to make use of such specific algorithms, and in fact typically show little difference between problems that are and are not formally NP-complete.

The vast majority of shells on typical beaches do not have especially elaborate patterns.

The result is that on page 464 the limiting form of the average behavior does not end up being an ordinary Gaussian.

And indeed it required solving innumerable small geometrical puzzles to do so.

From the early days of television in the 1950s, some attempts were made to do similar kinds of compression for images.