Shape

276 II Seeing How It Works
points. Still, even with approximations, I can calculate to any degree of accuracy I
wish.
Let’s begin with the line defined in the familiar equation
y ¼ mx þ b
and its segments determined by distinct endpoints. Here the quartet of points on the
line
are such that p < q < r < s. Moreover, the segments pq, pr, ps, qr, qs, and rs are
embedded in the line ps, as the semilattice
shows—a segment xy is embedded in a segment xAyA whenever xA a x and y a yA. But
this kind of ordering isn’t the only way to define embedding. There are metric devices,
as well. For example, a point is coincident with a line whenever the length of the line
is the sum of the two distances from the point to the endpoints of the line. Then, a
line l is embedded in another line lA if the endpoints of l are both coincident with lA.
When the embedding relation is defined, the part relation is, too. I need only maximal
lines. The reduction rules in table 4 are for sum, that is to say, maximal lines for any
set of lines. Similarly, the rules in table 5 produce maximal lines for difference. And
a transformation of a line is determined by the transformation of its endpoints. So
the result
ðC
tðAÞÞ þ tðBÞ
of applying a rule A fi B to a shape C is completely defined. Even better, I can find the
different transformations t that satisfy the formula
tðAÞ a C
using registration points and the conditions for determinate rules in table 9. What a
nice way for everything to turn out. There’s an algorithm—a zero-dimensional Turing
machine—for calculating with shapes of dimension greater than zero.
It doesn’t take all that much to get the job done for lines. In practice, however,
there are a host of complications to make sure that everything runs smoothly and efficiently.
And extending this to planes and solids, and exotic curves and surfaces, is not
without difficulty and real interest. Nonetheless, on the one hand, whatever I can do
with lines—visually—I can do with points or symbols. And then on the other hand,
calculating with points is a special case of calculating with lines, etc., when embedding