David Peat

44 From Certainty to Uncertaintyexample, branching and division. The complex pattern of tiles in amosque is the result of a basic pattern repeated over and over again.Related patterns are also found in Arabic music. Likewise the beautifulcrystal structures found in nature are the result of a repetitive processwhereby atoms take up positions next to their neighbors.Termite nests found in the tropics are several feet high and appearto be masterpieces of architectural construction. Yet no termite has inits head an overall plan of the nest. Rather, individual termites carryout extremely simple tasks of carrying particles of soil and placingthem in piles. Using a simple, repetitive rule the entire nest takes shape.There are endless examples of elaborate structures and apparentlycomplex processes being generated through simple repetitive rules, allof which can be easily simulated on a computer. It is therefore temptingto believe that, because many complex patterns can be generatedout of a simple algorithmic rule, all complexity is created in this way.Likewise, because fractals can reproduce the shapes of trees, rivers,clouds, and mountainsides it is seductive to believe that all naturalsystems grow and develop according to algorithmic fractal rules.Gödel’s theorem points to an essential limitation in this way of thinking.A great deal of complex behavior, but not everything, can be explainedthrough algorithms.Take, for example, what is known as Penrose tiling. Most systemsof laying down tiles—in other words of growing ever larger patternsthrough simple acts of repetition—require only a simple rule thatshows how one tile is to be placed next to its neighbor. Proceeding inthis way a person could lay down tiles all day without ever standing upto look at the overall effect. The mathematician Roger Penrose, however,pointed out that a very special system of tiling exists in which aneighborhood rule will never be sufficient to complete the pattern.Start laying down such tiles and sooner or later the next tile will fail tofit into the pattern. The only way Penrose tiles can be laid is by standingback and looking at the overall effect. Whereas algorithms workthrough local rules, Penrose tiles require an appreciation of the overallglobal plan.What’s more, certain crystals have been discovered that exhibitthe same sort of symmetry as Penrose’s tiles. This means that these