david_gelernter's picture
Computer Scientist, Yale University; Chief Scientist, Mirror Worlds Technologies; Author, America-Lite: How Imperial Academia Dismantled our Culture (and ushered in the Obamacrats)
Recursive Structure

Recursive structure is a simple idea (or shorthand abstraction) with surprising applications beyond science.

A structure is recursive if the shape of the whole recurs in the shape of the parts: for example, a circle formed of welded links that are circles themselves. Each circular link might itself be made of smaller circles, and in principle you could have an unbounded nest of circles made of circles made of circles.

The idea of recursive structure came into its own with the advent of computer science (that is, software science) in the 1950s. The hardest problem in software is controlling the tendency of software systems to grow incomprehensibly complex. Recursive structure helps convert impenetrable software rainforests into French gardens — still (potentially) vast and complicated, but much easier to traverse and understand than a jungle.

Benoit Mandelbrot famously recognized that some parts of nature show recursive structure of a sort: a typical coastline shows the same shape or pattern whether you look from six inches or sixty feet or six miles away.

But it also happens that recursive structure is fundamental to the history of architecture, especially to the gothic, renaissance and baroque architecture of Europe — covering roughly the 500 years between the 13th and 18th centuries. The strange case of "recursive architecture" shows us the damage one missing idea can create. It suggests also how hard it is to talk across the cultural Berlin Wall that separates science and art. And the recurrence of this phenomenon in art and nature underlines an important aspect of the human sense of beauty.

The re-use of one basic shape on several scales is fundamental to medieval architecture. But, lacking the idea (and the term) "recursive structure," art historians are forced to improvise ad hoc descriptions each time they need one. This hodgepodge of improvised descriptions makes it hard, in turn, to grasp how widespread recursive structure really is. And naturally, historians of post-medieval art invent their own descriptions—thus obfuscating a fascinating connection between two mutually alien aesthetic worlds.

For example: One of the most important aspects of mature gothic design is tracery — the thin, curvy, carved stone partitions that divide one window into many smaller panes. Recursion is basic to the art of tracery.

Tracery was invented at the cathedral of Reims circa 1220, and used soon after at the cathedral of Amiens. (Along with Chartres, these two spectacular and profound buildings define the High Gothic style.) To move from the characteristic tracery design of Reims to that of Amiens, just add recursion. At Reims, the basic design is a pointed arch with a circle inside; the circle is supported on two smaller arches. At Amiens, the basic design is the same — except that now, the window recurs in miniature inside each smaller arch. (Inside each smaller arch is a still-smaller circle supported on still-smaller arches.)

In the great east window at Lincoln Cathedral, the recursive nest goes one step deeper. This window is a pointed arch with a circle inside; the circle is supported on two smaller arches — much like Amiens. Within each smaller arch is a circle supported on two still-smaller arches. Within each still-smaller arch, a circle is supported on even-smaller arches.

There are other recursive structures throughout medieval art.

Jean Bony and Erwin Panofsky were two eminent 20th century art historians. Naturally they both noticed recursive structure. But neither man understood the idea in itself. And so, instead of writing that the windows of Saint-Denis show recursive structure, Bony said that they are "composed of a series of similar forms progressively subdivided in increasing numbers and decreasing sizes." Describing the same phenomenon in a different building, Panofsky writes of the "principle of progressive divisibility (or, to look at it the other way, multiplicability)." Panofsky's "principle of progressive divisiblity" is a fuzzy, roundabout way of saying "recursive structure."

Louis Grodecki noticed the same phenomenon—a chapel containing a display-platform shaped like the chapel in miniature, holding a shrine shaped like the chapel in extra-miniature. And he wrote that "This is a common principle of Gothic art." But he doesn't say what the principle is; he doesn't describe it in general or give it a name. William Worringer, too, had noticed recursive structure. He described gothic design as "a world which repeats in miniature, but with the same means, the expression of the whole."

So each historian makes up his own name and description for the same basic idea—which makes it hard to notice that all four descriptions actually describe the same thing. Recursive structure is a basic principle of medieval design; but this simple statement is hard to say or even think if we don't know what "recursive structure" is.

If the literature makes it hard to grasp the importance of recursive structure in medieval art, it's even harder to notice that exactly the same principle recurs in the radically different world of Italian Renaissance design.

George Hersey wrote astutely of Bramante's design (ca 1500) for St Peter's in the Vatican that it consists of "a single macrochapel…, four sets of what I will call maxichapels, sixteen minichapels, and thirty-two microchapels." "The principle [he explains] is that of Chinese boxes — or, for that matter, fractals."

If he had only been able to say that "recursive structure is fundamental to Bramante's thought," the whole discussion would have been simpler and clearer — and an intriguing connection between medieval and renaissance design would have been obvious.

Using instead of ignoring the idea of recursive structure would have had other advantages too.

It helps us understand the connections between art and technology; helps us see the aesthetic principles that guide the best engineers and technologists, and the ideas of clarity and elegance that underlie every kind of successful design. These ideas have practical implications. For one, technologists must study and understand elegance and beauty as design goals; any serious technology education must include art history. And we reflect, also, on the connection between great art and great technology on the one hand and natural science on the other.

But without the right intellectual tool for the job, new instances of recursive structure make the world more complicated instead of simpler and more beautiful.