Shape

184 II Seeing How It Works
Embedding is given for basic elements, and the part relation is defined for the
shapes they make. The generalization looks like this
parts : shapes < embedded elements : basic elements
and the supporting definition is remarkably easy to state—it’s almost automatic—
thanks to maximal elements.
One shape is part of another shape whenever every maximal element of the first is
embedded in a maximal element of the second.
Thus for example, the lowercase k
with three maximal lines is part of the shape
with eight maximal lines, as this trio of embeddings shows
Parts don’t require identity. The maximal lines in the two shapes are completely different.
And it’s worth making a big deal of this—
parts are there whenever I can trace them out
—so that the formality doesn’t obscure the result. Nothing is lost in this two-tiered
approach. There are basic elements and arrangements of them, allowing the former to
keep all of their properties in the latter. Shapes are combinatorial after all, but not in
the usual way where the properties of sets, list structures, graphs, etc., determine how
elements behave. Formal devices come at the end to tidy things up. They don’t make
shapes the way they are. In a word, that’s preposterous—at least as Jacques Barzun
means it.
It’s easy to see that the part relation is a partial order on shapes. In fact, I can
use it to define lattices for shapes in the way I used embedding to describe basic elements,
only now with far more generality. Meets and joins are defined for all shapes
and for all values of i. This works because shapes can contain multiple elements and
not just single ones, and also because there’s an empty shape with no elements at