Notes

Chapter 2: The Crucial Experiment

Section 3: Why These Discoveries Were Not Made Before


Ornamental art

Almost all major cultural periods are associated with certain characteristic forms of ornament. Often the forms of ornament used on particular kinds of objects probably arose as idealized imitations of earlier or more natural forms for such objects—so that, for example, imitations of weaving, bricks and various plant forms are common. Large-scale purely abstract patterns were also central to art in such cultural traditions as Islam where natural forms were considered works of God that must not be shown directly. Once established, styles of ornament tend to be repeated extensively as a way of providing certain comfort and familiarity—especially in architecture. The vast majority of elaborate ornament seems to have been created by artisans with little or no formal theoretical discussion, although particularly since the 1800s there have been various attempts to find systematic ways to catalog forms of ornament, sometimes based on analogies with grammar. (Issues of proportion have however long been the subject of considerable theoretical discussion.) It is notable that whereas repetitive patterns have been used extensively in ornament, even nesting is rather rare. And even though for example elaborate symmetry rules have been devised, nothing like cellular automaton rules appear to have ever arisen. The results in this book now show that such rules can capture the essence of many complex processes that occur in nature—so that even though they lack historical context such rules can potentially provide a basis for forms of ornament that are familiar as idealizations of nature. (Compare page 929.)

The pictures in the main text show a sequence of early examples of various characteristic forms of ornament.

22,000 BC (Paleolithic). Mammoth ivory bracelet from Mezin, Ukraine. Similar zig-zag designs are seen in other objects from the same period. In the example shown, it is notable that the angle of the zig-zags is comparable to the angle of the Schreger lines that occur naturally in mammoth dentin.

3000 BC (Sumerian). Columns with three colors of clay pegs set in mud from a wall of the Eanna temple in Uruk, Mesopotamia (Warka, Iraq)—perhaps mentioned in the Epic of Gilgamesh. (Now in the Staatliche Museum, Berlin.) This is the earliest known explicit example of mosaic.

1200 BC (Greek). The back of a clay accounting tablet from Pylos, Greece. The pattern was presumably made by the procedure shown below. Legend has it that it was the plan for the labyrinth housing the minotaur in the palace at Knossos, Crete, and that it was designed by Daedalus. It is also said that it was a logo for the city of Troy—or perhaps the plan of some of its walls. The pattern—in either its square or rounded form—has appeared with remarkably little variation in a huge variety of places all over the world—from Cretan coins, to graffiti at Pompeii, to the floor of the cathedral at Chartres, to carvings in Peru, to logos for aboriginal tribes. For probably three thousand years, it has been the single most common design used for mazes.

900 BC (Phoenician). Ivory carving presumably from the Mediterranean area. (Now in the British Museum.) This was a common decorative pattern, formed by drawing circles centered at holes arranged in a triangular array. It is also found in Egyptian and other art. Such patterns were discussed by Euclid and later Leonardo da Vinci in connection with the theory of lunes.

1^st century BC (Celtic). The back of the so-called Desborough Mirror—a bronze mirror from Desborough, England made in the Iron Age sometime between 50 BC and 50 AD. (Now in the British Museum.) The engraved pattern is made of parts of circles that just touch each other, as in the picture below.

2^nd century AD (Roman). A mosaic from a complex in Rome, Italy. (Now in the National Museum, Rome.) The geometrical pattern was presumably made by first constructing 48 regularly spaced spokes by repeated angle bisection, as in the first picture below, then drawing semicircles centered at the end of each spoke, and finally adding concentric circles through the intersection points. Similar rosette patterns may have been used in Greece around 350 BC; they became popular in churches in the 1500s.

8^th century (Islamic). A detail on the outside wall of the Great Mosque of Córdoba, Spain, built around 785 AD.

8^th century (Celtic). An area less than 2 inches square from inside the letter ρ on the extremely elaborate chi-rho page of the Book of Kells, an illuminated gospel manuscript created over a period of years at various monasteries, probably starting around 800 AD at the Irish monastery on the island of Iona, Scotland. Even on this one page there are perhaps a dozen other very similar nested structures.

12^th century (Italian). A window in the Palatine Chapel in Palermo, Sicily, presumably built around 1140 AD. The chapel is characteristic of so-called Arab-Norman style.

13^th century (English). The Dean's Eye rose window of the Lincoln cathedral in England, built around 1225 AD. Similar tree-like patterns are seen in many Gothic windows from the same general period.

13^th century (Italian) (4 pictures). Marble mosaics on the floor of the cathedral at Anagni, Italy, made around 1226 AD by Cosmas of the Cosmati group. (The fourth picture is a close-up of the third.) The third picture—particularly the part magnified in the fourth picture—shows an approximate nested structure, presumably created as in the pictures below. The triangles are all equilateral, with the result that at a given step several different sizes of triangles occur—though the basic structure of the pattern is still the same as from the rule 90 cellular automaton. (Compare the Apollonian packing of page 986.) The Cosmati group—mostly four generations of one family—made elaborate geometrical and other mosaics with a mixture of Byzantine, Islamic and other influences from about 1190 to 1300, mostly in and around Rome, but also for example in Westminster Abbey in England. Triangular shapes with one level of nesting are quite common in their work; three levels of nesting as shown here are rare. It is notable that in later imitations of Cosmati mosaics, these kinds of patterns were almost never used.

14^th century (Islamic). Wall decoration in the Pir-i-Bakran mausoleum in Linjan, Iran, built around 1299-1312. The pattern is square Kufi calligraphy for a widely quoted verse of the Koran. The pictures below illustrate its relation to standard cursive Arabic writing. The basic Kufi style was already in use as a variant by 700 AD—with the square form being seen in architectural ornament by about 1100 AD.

14^th century (Islamic). Tiled wall in the Alcázar of Seville, Spain, built in 1364. (The same pattern was used at about the same time in the Alhambra in Granada, Spain.) The pattern can be made by starting with a grid of triangles, then consistently pushing in or out the sides of each one. (Notable uses of such patterns were made by Maurits Escher starting in the 1930s.)

Other cases. The cases that are known inevitably tend to be ones created out of stone or ceramic materials that survive; no doubt there were others created for example with wood or textiles. One case with wood is Chinese lattice. What has survived mostly shows repetitive patterns, but the ice-ray style, probably going back to 100 AD, has approximate nesting, though with many random elements. The patterns shown are all basically two-dimensional. An example of 1D ornamental patterns are molding profiles. Ever since antiquity these have often been quite elaborate, and it is conceivable that they can sometimes be interpreted as showing nesting.


From Stephen Wolfram: A New Kind of Science [citation]