Shape

196 II Seeing How It Works
Table 7
Some Properties of Shapes
Algebra
Basic
elements
Boundary
shapes
Number of
parts
U 0j points none finite
U 1j lines U 0j indefinite
U 2j planes U 1j indefinite
U 3j solids U 2j indefinite
Table 8
More Properties of the Part Relation
U ij
Every shape has a distinct
nonempty part—there is
no smallest shape
Every shape is a distinct part
of another shape—there is
no largest shape
0 ¼ i ¼ j no no
0 ¼ i < j no yes
0 < i a j yes yes
The description of parts in table 7 misses some important details. These require
the index j. The additional relationships are given in table 8.
Boundaries of Shapes Are Shapes
Boundaries of shapes are another way to show how basic elements combine, and how
embedding works. If a shape has a boundary, then its boundary is a shape. The boundary
of each maximal element in the shape is a shape, and the sum of these shapes is
the boundary of the shape.
Shapes containing points don’t have boundaries. But the boundary of a shape
made up of lines, planes, or solids is a shape containing points, lines, or planes.
(The empty shape—when it’s used with shapes with boundaries—is the only shape
that has an empty boundary.) Some shapes and their boundaries are shown in these
examples—first for lines