Shape

291 Erasing and Identity
to produce the third decomposition. (Like results follow in the algebra U 12 þ W 22 with
the schema bðxÞ fi x for lines and the shapes they bound. I can do the third decomposition
exactly as before in six steps or, alternatively, in two steps. The fewest steps for
each decomposition is the number of gray tones it contains. Boundaries have some
nice uses.) As I’m going to show next—this is the second point of interest—the number
of decompositions for each of the shapes in the series
is fixed in a definite way, but how these decompositions are represented as shapes can
vary freely. Decompositions are abstract in the same way words and sentences are in
generative grammars, and they’re abstract in the same way as Turing machines, cellular
automata, etc. How I show them is independent of what they do—it’s a question of
design. Playing around with algebras and rules is an effective means to explore the
possibilities.
The number f ðnÞ of distinct decompositions for any shape in the series
when the shape is divided into squares and triangles can be given in terms of its
numerical position n. The first shape in the series is described in one way—it’s a
square—and the second shape can be described in two ways—it’s either two squares
or four triangles. Moreover, the number of possible descriptions for each of the succeeding
shapes—so long as I keep the remainder empty—is the corresponding term in
the series of Fibonacci numbers defined by f ðnÞ ¼ f ðn 1Þþ f ðn 2Þ. This works in
the following way