Structure is at the rock bottom of all explanatory sciences: the intriguing patterns of symmetry and antisymmetry, part-whole relations and rhythmic repetition that not only fascinate the mathematician and the physicist, but also the musician, the artist and the linguist. Structural notions such as symmetry are unifying concepts and at the core of many forms of scientific and artistic endeavor. For me, there is no great difference between the structures found in scientific inquiry and those encountered in the arts, particularly the decorative arts, music and architecture. Sometimes, structures in the arts are connected with emotion, but that is by no means necessary. Obviously, the various arts differ among themselves in that respect. Like many people, I experience strong emotions sometimes while enjoying the beauty of music, but I am never brought to tears by tile decorations or even most architecture, no matter how rich, fascinating and beautiful the latter can be.

On the other hand, the process of scientific discovery, ultimately the discovery of a unifying structural pattern underlying a variety of phenomena, is not without emotional responses of its own. What makes the mathematically oriented sciences so exciting is that the patterns discovered are not given by common sense. The patterns are just hidden, sometimes to an extreme degree, and figuring out the underlying order of things is one of the great adventures one can embark upon in life. 

It is not yet generally known, but theoretical linguistics has been fairly successful in the second half of the 20th century in unraveling the hidden patterns underlying the thousands of natural languages of the world. We don't just utter strings of words, but we organize our words into larger units in an intricate way which is typical of our kind. These patterns not only show hierarchical organization, but also recursion, which means that we unconsciously create structures with parts that have the same form as the whole containing them.

Mathematically speaking, such structures --which we produce almost every minute-- are members of a whole family of self-similar structures such as fractals and other iterated function systems. Such structures can be visualized in often spectacular ways and play a big role in computer art.  As a matter of fact, more traditional forms of art, like architecture and decoration, are full of self-similar patterns, too.

Our language, in other words, expresses hidden mathematical patterns of a kind that we also articulate  --in a less unconscious way-- in the arts. An example of the latter are the patterns studied by Marius Hardonk is his book Cross-Cultural Universals of Aesthetic Appreciation in Decorative Band Patterns. Nijmegen: Nijmegen Institute for Cognition and Information, 1999.

Humans all over the world, far from being endlessly malleable as postmodern fashion claims, often express themselves in universal mathematical patterns of surprising beauty (see, for instance, Marcia Ascher, Ethnomathematics. Boca Raton etc.: Chapman and Hall/CRC, 1998).

 

Last update: 11 March 2000