Shape

221 ‘‘Nor Do Lineaments Have Anything to Do with Material’’
section—and so will only give a brief outline. In fact, I’ll just do adding and subtracting,
as each normally implies the part relation (table 6, facts 1 and 2). The idea is to
insert an algebra of weights—with labels, it’s just the Boolean algebra for sets—in an
algebra of shapes. Points are trivial. The problem is with lines, planes, and solids.
The basic elements in a weighted shape are maximal in the sense that ones of
equal weight are. For a weighted shape A, let A be the shape defined when all of its
weights are stripped away. Let’s suppose A is the weighted shape
Then A is the shape
And for planes, if A is
then A is
The sum of two weighted shapes A and B in an algebra W ij is formed by dividing
the basic elements in A and B into separate pieces to assign weights. These new basic
elements are distinguished according to whether they contribute to the differences
A B or B A , or to the product A B . These parts are distinct and separate from
one another, and they totally exhaust the sum A þ B
The weights assigned to the basic elements that contribute to the differences are taken
directly from A and B, while the weights assigned to the basic elements that contribute
to the product are defined in unions. This process is easy to illustrate for lines