Shape

226 II Seeing How It Works
intricate examples as well—a few easy ones are given in this part and others in part III.
Right now, though, it’s good to see that my definition is more than bookkeeping. Not
only does it imply that things connect up in designs in alternative ways, it also implies
that specialists and users can communicate across domains—even at cross-purposes—
to contribute to a single end. This is cooperation without coercion. There’s no need
for shared understanding or overarching control. Calculating with ambiguity and
changing connections makes this possible. And it shows the kind of virtuosity that’s
expected in intelligent and creative practice. This goes beyond familiar slogans, like
‘‘form follows function,’’ that are supposed to guide practice, and tiresome theory.
And it shows the inadequacy of presumed generators of designs such as programme,
context, technology, and material, and of studied accounts of how designs are produced
whether formal, functional, rational, or historical. Practice is far more interesting.
Designs result from a confluence of activities with multiple perspectives
that ebb and flow. Changing interests and goals interact and influence one another
dynamically. Nothing is set for long in this process. Designs are complicated and
multifaceted—they’re the very stuff of algebras and calculating.
Solids, Fractals, and Other Zero-Dimensional Things
I said earlier that decompositions are finite sets of parts that sum to make shapes. They
show how shapes are divided for different reasons and how these divisions interact. Of
course, decompositions aren’t defined only for shapes in an algebra U ij . More generally,
they’re defined for whatever there is in any algebra formed from the algebras in
the series U ij . And decompositions have their own algebras. Together with the empty
set—it’s not a decomposition—they form generalized Boolean algebras with operators.
There’s the subset relation, the standard operations for sets (union, intersection, and
relative complement), and the Euclidean transformations. And, in fact, these are just
the kind of algebras needed for set grammars. They make decompositions zero dimensional
like the shapes in the algebras U 0 j . The parts in decompositions behave exactly
like points. Whatever else they are, they’re members of sets—units that are independent
in combination and without finer analysis.
I like to point out that complex things like fractals and computer models of solids
and thought are zero dimensional. Am I obviously mistaken? It’s easy to think so. Solids
are clearly three dimensional—I bump into them. Fractal dimensions are neither
whole numbers nor zero, whatever they are. And no one knows about thought, even
though it’s said to be multidimensional when it’s creative. Only this misses the point.
Fractals and computer models rely on decompositions that are zero dimensional, or
like representations such as lists or graphs in which units are also given from the start.
That’s how computers work—they manipulate complex things with predefined divisions.
Units are easy to count—fractal dimensions add up before calculating begins—
and easy to move around to sift through possible configurations for ones of interest.
Nonetheless, computers fail with shapes once they include lines or basic elements of
higher dimension. I can describe what happens to shapes as long as I continue to cal-