Shape

260 II Seeing How It Works
Now, no segment of an oblique line is long enough for the rule
to apply. Only the ambiguity needn’t end here if I go on calculating. I may have to
look farther out than I can see. Until I erase the horizontal line, the point can interact
with any line that’s defined—if there’s a rule for it. Still, my answer may not matter.
What if I also have the rule
that adds to lines. Dividing shapes into independent pieces to define rules rarely succeeds,
unless I appeal to labels or comparable devices—for example, compound shapes
in algebras formed in direct products—to keep things separate in an artificial way. But
then what I see may not be what’s there. Calculating with shapes is different when surprises
are unavoidable.
There’s a lot to see as I use rules to calculate with shapes. Everything works with
everything else. But I’ve been showing only easy examples, special cases, and clever
tricks. What can I say about rules generally to help me understand what they do? It’s
all in the two formulas for applying rules—either in the transformation t or in the part
relation a.
Classifying Rules with Transformations
Rules can be classified in a variety of important ways. First, it’s possible to decide
whether they apply determinately or indeterminately—that is to say, whether or not
there are a limited number of transformations t that satisfy the formula
tðAÞ a C
for the rule A fi B and the shape C. The conditions for determinate rules vary somewhat
from algebra to algebra, but can be framed in terms of a recursive taxonomy of
registration marks defined with basic elements. It’s usually a surprise that these marks
aren’t always the basic elements themselves or their boundary elements. Paradigmatically,
lines in shapes intersect at points