Shape

261 Classifying Rules with Transformations
Table 9
Determinate Rules in the Algebras U 1j
Algebra
U 11
U 12
U 13
The rule A fi B is determinate
Never.
Three lines in A do not intersect at a common point. Further, no two of these lines
are collinear, and all three are not parallel.
There are two cases. (1) Two lines in A are skew. (2) Three lines in A do not intersect
at a common point. Further, no two of these lines are collinear, and all three are not
parallel.
to fix transformations. The conditions that make rules determinate in the algebras U 1 j
are specified fully in table 9 to show how this works. When j is 1, there are no registration
points—lines are always collinear. Otherwise, when j is 2 or j is 3, there are at
least two such points and a plane. (What happens when i is 0?)
Many examples of determinate rules are evident in the previous section—and
some contain more than lines. Of interest, all the rules made up of points and lines
are determinate. But indeterminate rules are also easy to find, with and without points.
The identity
is indeterminate in the algebra U 12 . The K has three maximal lines that intersect at a
single point. So the identity applies under indefinitely many changes in scale to find
K’s in the shape
However, bringing in a point
may not help. There’s still just one registration point—the point and the intersection
of the lines in the K are the same. Moreover, notice that different shapes may be equivalent
in terms of the registration marks they define. The lines in the uppercase K and
the lines in the lowercase k intersect at the same point, but this isn’t all. Shapes made