The concept of rules

Processes based on rules occur in a great many areas of human endeavor. Sometimes the rules serve mainly as a constraint. But it is not uncommon for them to be used—like in a cellular automaton—as a way of specifying how structures should be built up. Almost without exception, however, the rules have in the past been chosen to yield only rather specific and simple results. Beyond ornamental art, examples with long histories include:

Architecture. Structures such as ziggurats and pyramids were presumably constructed by assembling collections of stones according to simple rules. The Great Pyramid in Egypt was built around 2500 BC and contains about two million large stones. (By comparison, the pictures of rule 30 on pages 29 and 30 contain a total of about a million cells.) Starting perhaps as long ago as 1000 BC Hindu temples were constructed with similar elements on different scales, yielding a form of approximate nesting. In Roman and later architecture, rooms in buildings have quite often been arranged in roughly nested patterns (an extreme example being the Castel del Monte from the 1200s). From the Middle Ages many Persian gardens (such as those of the Taj Mahal from around 1650) have had fairly regular nested structures obtained by a few successive fourfold subdivisions. And starting in the early 1200s, Gothic windows were often constructed with levels of roughly tree-like nested forms (see above). Nesting does not appear to have been used in physical city plans (except to a small extent in Vauban star fortifications), though it is common in organizational structures. (As indicated above, architectural ornament has also often in effect been constructed using definite rules.)

Textile making. Since early in human history there appear to have been definite rules used for weaving. But insofar as the purpose is to produce fabric the basic arrangement of threads is normally always repetitive.

Rope. Since at least 3000 BC rope has been made by twisting together strands themselves made by twisting, yielding cross-sections with some nesting, as in the second picture below. (Since the development of wire rope in the 1870s precise designs have been used, including at least recently the 7×7×7 one shown last below.)

Knots and string figures. For many thousands of years definite rules have been used for tying knots and presumably also for making string figures. But when the rules have more than a few steps they tend to be repetitive.

Paperfolding. Although paperfolding has presumably been practiced for at least 2000 years, even the nested form on page 892 seems to have been noticed only very recently.

Mathematics. Ever since Babylonian times arithmetic has been done by repeatedly applying simple rules to digits in numbers. And ever since ancient Greek times iterative methods have been used to construct geometrical figures. In the late 1600s the idea also emerged that mathematical proofs could be thought of as consisting of repeated applications of definite rules. But the idea of studying possible simple rules independent of their purpose in generating results seems never to have arisen. And as mathematics began to focus on continuous systems the notion of enumerating possible rules became progressively more difficult.

Logic. Rules of logic have been used since around 400 BC. But beyond forms like syllogisms little seems to have been studied in the way of generating identifiable patterns from them. (See page 1099.)

Grammar. The idea that human language is constructed from words according to definite grammatical rules has existed since at least around perhaps 500 BC when Panini gave a grammar for Sanskrit. (Less formal versions of the idea were also common in ancient Greek times.) But for the most part it was not until about the 1950s that rules of grammar began to be viewed as specifications for generating structures, rather than just constraints. (See page 1103.)

Poetry. Definite rules for rhythm in poetry were already well developed in antiquity—and by perhaps 200 BC Indian work on enumerating their possible forms appears to have led to both Pascal's triangle and Fibonacci numbers. Patterns of rhyme involving iterated length-6 permutations (sestina) and interleaved repetitive sequences (terza rima) were in use by the 1300s, notably by Dante.

Music. Simple progressions and various forms of repetition have presumably been used in music since at least the time of Pythagoras. Beginning in the 1200s more complex forms of interleaving such as those of canons have occasionally been used. And in the past century a few composers have implicitly or explicitly used structures based on simple Fibonacci and other substitution systems. Note that rules such as those of counterpoint are used mainly as constraints, not as ways of generating structure.

Military drill. The notion of using definite rules to organize and maneuver formations of soldiers appears to have existed in Babylonian and Assyrian times, and to be well codified by Roman times. Fairly elaborate cases were described for example by Niccolò Machiavelli in 1521, but all were set up to yield only rather simple behavior, such as a column of soldiers being rearranged into lines. (See the firing squad problem on page 1035.)

Games. Games are normally based on definite rules, but are set up so that at each step they involve choosing one of many possibilities, either by skill or randomness. The game of Go, which originated before 500 BC and perhaps as early as 2300 BC, is a case where particularly simple rules manage to allow remarkably complex patterns of play to occur. (Go involves putting black and white stones on a grid, making it visually similar to a cellular automaton.)

Puzzles. Geometric and arithmetic puzzles surprisingly close to those common today seem to have existed since as long ago as 2000 BC. Usually they are based on constraints, and occasionally they can be thought of as providing evidence that simple constraints can have complicated solutions.

Cryptography. Rules for encrypting messages have been used since perhaps 2000 BC, with non-trivial repetitive schemes becoming common in the 1500s, but more complex schemes not appearing until well into the 1900s. (See page 1085.)

Maze designs. From antiquity until about the 1500s the majority of mazes followed a small number of designs—most often based directly on the one shown on page 873, or with subunits like it. (It is now known that there are many other designs that are also possible.)

Rule-based pictures. It is rather common for geometric doodles to be based on definite rules, but it is rare for the rules to be carried far, or for the doodles to be preserved. Some of Leonardo da Vinci's planned book on "Geometrical Play" from the early 1500s has, however, survived, and shows elaborate patterns satisfying particular constraints. Various attempts to enumerate all possible patterns of particular simple kinds have been made—a notable example being Sébastien Truchet in 1704 drawing 2D patterns formed by combining , , , in various possible ways.