The simple diagram at the left illustrates how the "actual"
formative relationships in geometry are often times based on some
higher, more perfect "ideals". This schematic depicts the "actual"
relationship between both a unit-volume sphere and cube; and, a unit-
surface sphere and cube.

The smaller of the two squares represents the face of the unit-
surface cube set upon the X-section of the unit-surface sphere (the
circle). The larger square is the face of the unit-volume cube with the
(same) circle representing this time a unit-volume sphere. In both
relationships the forms of the respective cube and sphere pairs are seen
to be "nearly" commensurate . . . but not "perfectly" so. This is because
tiny tips of the small and large squares protrude very slightly past the
circle's circumference.

The schematic on the right is illustrating "perfect"
commensuration. We could say that this arrangement is the
ideal that is patterning the actual relationships depicted on the
left. In this case, the edges of the cubes share common points on
the circumference of their respective spheres. Again, this is the
"ideal" pattern to which the "actual" relationships can only
hope to approach.

On inspection, and against a backdrop of the whole of The
Geometry Of Form, this slight "imperfection" in what would
otherwise be perfect formal coordination between the cube and
sphere, and the geometric qualities of surface and volume,
reveals itself as anything but a chaotic aberration, or
"coincidental" arrangement. For the mathematics defining this
variance carries throughout The Geometry Of Form and derives
from primal geometric transformations.