Shape

279 I Don’t Like Rules—They’re Too Rigid
Easy enough—but the generalization presents a problem if all I have is rules. Squares
work because they’re rigid—hence the title of this section—while relationships between
lines and angles can vary arbitrarily in quadrilaterals. Quadrilaterals inscribed
one in another needn’t repeat the same spatial relation. There’s plenty of room for
things to vary. And this is also true for squares—perhaps in the first black and white
drawing above, but explicitly here
This kind of variation seems to imply that I need an indefinite number of rules to
correspond with different spatial relations for squares. There are too many ways they
can go together. But there’s a neat way out using alternative spatial relations for points
and lines—endpoints and interior points. I can modify my original rule
by replacing the square in its left side and the new square in its right side with their
boundaries to get the rule
And this rule gives me exactly what I want, when I use it together with the indeterminate
rule
that moves a point anywhere in the interior of a line, à la Zeno, half a segment at a
time, and the rule
that erases a point. Then the boundaries of squares can be used to inscribe squares in
squares, even as these boundaries are moved from place to place one point at a time.
The details of this process are illustrated in the following way—starting with the topmost
point—to orient squares as desired in a kind of distributed calculating.