Shape

200 II Seeing How It Works
maximal elements in the shapes are still so when they’re combined. The rules in table
4 that define maximal lines show why. This is perfectly general—the same goes for
planes and solids—and it’s easy to see. Let’s begin with the second case in the first
rule in table 4 for one line embedded in another
and the second rule for two overlapping lines with a common segment not identical to
either
In both, the boundary of the sum of the lines—that’s the boundary of the maximal
line they define—
and the boundary of their common segment
combine to form the sum of the boundaries of the lines
In symbols, it’s
bðl þ lAÞþbðl lAÞ ¼bðlÞþbðlAÞ
(I’m using l and lA in two ways to denote lines and the shapes they define. I can be
rigorous with added symbols—the shapes are flg and flAg—but it’s too fussy.) The third
rule in table 4 shows another relationship. Two lines are discrete with a boundary element
in common
The boundary of their sum—again, it’s the boundary of the maximal line containing
both—
and the product of their boundaries—their common endpoint—
come together in the sum of the boundaries of the lines
And in symbols, it’s