Shape

224 II Seeing How It Works
suggest an interesting rule. Weights form a lattice in which the least upper bound of
any two is their sum, and the greatest lower bound is their product or the minimum
of both
u v ¼ minfu; vg
Clearly, u a v when
u v ¼ u
But in full Boolean fashion, I should also have the equality
u v ¼ u ðu vÞ
in which the product of u and v is determined in a double difference. If I make this
so, then the rule where the difference of u and v is their arithmetic difference when
v is less than u, and zero otherwise, is a felicitous choice. All of it—parts, sums, and
differences—looks like this
Weights in the way I’ve been describing them have physical properties, so the
transformations might make a difference of another kind. A color here needn’t be
the same when it’s put over there, and area, thickness, and tone can vary as I move
weighted shapes around. Nonetheless, I’ll do what’s easy and simply avoid additional
complication—weights are invariant under the transformations. This makes perfectly
good sense, too, being partly confirmed in everyday experience. My weight doesn’t
change when I go for a walk. No matter how I move, it’s exactly the same.
Of course, it’s always possible to elaborate the algebras U ij in other ways that go
beyond labels and weights. I can use labels and weights together, and it’s a nice exercise
to see how this sets up algebraically. But there’s no reason to be parsimonious—
and no elegance is lost—when new algebras are this easy to define. The idea is to get
the algebra that makes sense for what you’re doing, and not to make what you’re
doing conform to an arbitrary algebra that’s in use. The mathematics is generous—
even profligate—and it’s no sin to be prodigal. It’s worth taking the time to get the
right stuff to express what you want to when you calculate. That’s what I do when I
select appropriate materials—pencils, pens, markers, different kinds of paper, etc.—to
draw without thinking about how it’s calculating. But there’s mathematics for all of it.
And I can even imagine doing it on the fly, so that algebras change dynamically as
shapes do when rules are tried.
The algebras U ij , V ij , and W ij can also be combined in a variety of ways, for example,
in appropriate sums and products (direct products), to obtain new algebras.
These algebras typically contain compound shapes with one or more components in