Shape

186 II Seeing How It Works
both of the shapes. In fact, for shapes with points, sum is set union—it contains
exactly the points in one shape or the other. But, for other shapes without points,
maximal elements can fuse when they’re not identical—always if they overlap, and
sometimes if they’re discrete—and so may not be preserved when they’re combined.
The shape
is produced in the addition of the shape
made up of four triangles, and the shape with the two pieces
Neither of the shapes in the sum has any maximal element in common with the other.
Their maximal elements fuse in combination. And the parts of the sum—for example,
the shape
containing four triangles—needn’t combine the maximal elements in the shapes. Set
union is all there is to addition for shapes with points, but it’s a different story with
lines, planes, and solids when basic elements fuse and divide.
The way I’ve defined sum has a certain charm, but it isn’t constructive. Unless
I’m using points, I can’t always list the maximal elements in a sum in an easy
way. For this, I need an algorithm. I’m not going to give one for basic elements in
general—it’s too cumbersome. Rather, I’ll give an easy algorithm for lines. It shows
everything that’s involved, and how it all depends on embedding. The three reduction
rules in table 4 are used to produce maximal lines. The rules are independent, and they
exhaust the different ways lines fuse. They apply recursively in any order—but with no