Shape

262 II Seeing How It Works
up of segments of two or more unbounded lines intersecting at any common point—
for example
—are likewise related. Of course, with another registration mark, definite transformations
can be defined, as in the identity
This works in the same way for the identity
And both identities are equivalent with respect to their registration marks, but just the
first finds parts in the shape
The part relation is needed, too, to tell whether rules apply or not.
The conditions for determinate rules in table 9 depend on the properties of the
shapes in the left sides of rules, and not on the shapes to which the rules apply. This
keeps the classification the same no matter how I calculate. It’s also important to note
that the conditions in table 9 provide the core of an algorithm that enumerates the
transformations under which any determinate rule A fi B applies to a given shape C.
If C is A, then the algorithm defines the symmetry group of A. But this is only an aside.
More to the point, rules can be tried automatically.
The problem of finding transformations to apply a rule A fi B to a shape C looks
hard because embedding rather than identity may determine how maximal elements
correspond. Nonetheless, maximal elements in the shapes A and C fix registration
marks that coincide whenever a transformation makes A part of C. Certain combinations
of these marks are enough to scale A and align it with C. Alignments needn’t satisfy
the part relation—distinct shapes may have equivalent registration marks. But the