Shape

337 Chinese Lattice Designs—Seeing What You Do
the process. In fact, the process isn’t hard to describe. When I first thought about making
ice-rays, I had Dye and the schema x fi divðxÞ in mind. This is what I said—
One can imagine a Chinese artisan, summoned to a building site, bringing with him tools and
implements and a collection of finely finished sticks. Shown a rectangular window frame, he is
asked to create an ice-ray lattice. He begins his design by selecting a stick of the appropriate length
and carefully attaching it between two edges of the existing rectangular frame, thus forming two
quadrilateral regions. He continues his work by subdividing one of these areas into a triangle and
a pentagon. He further divides the triangle into a triangle and a quadrilateral; he divides the pentagon
into a quadrilateral and a pentagon. Each subdivision is made in the same way: attach an
appropriately sized stick between two edges of a previously constructed triangle or quadrilateral
or pentagon, so that it does not cross previously inserted pieces. Each stage of the construction is
stable; each stage follows the same rules.
Ice-rays make it easy to be a ‘‘rationalist’’—to believe that available technologies,
properties of materials, and methods of construction determine designs. And, to some
extent, these constraints do matter as they’re expressed in schemas and rules. Rules allow
for freedom and constraints—going back and forth from one to the other is always
possible. In a way, though, it’s ironic. Many times, the truly hard problems are to fix
constraints that aren’t immediately spatial, for example, so that rules apply in the right
logical or temporal sequence. But the seemingly effortless switch from constraints to
freedom via identities and other rules is more impressive. That’s where embedding
and transformations really make a difference.
I’ve tacitly assumed all along that rules are nondeterministic, that is to say, they
can be applied in various sequences under alternative transformations. This is the
source of creativity, in Chomsky’s sense, for vocabulary and syntax when different
things are produced combining the same constituents, and there’s a kind of ambiguity,
too, when the same thing results in different ways. But shapes and rules give much
more. Creativity and ambiguity have an Aristotelian origin, as well, that’s evident
when I use the schema x fi x to define identities. Then I can divide shapes to see as I
please.
For lattice designs, there are unlimited opportunities for variety and novelty in a
traditional style. Even so, rules may apply in more specific ways. For example, not
counting the pair of leftmost divisions, the ice-ray
is produced from left to right