Shape

277 I Don’t Like Rules—They’re Too Rigid
is identity. Symbolic calculating and visual calculating—counting and seeing—are the
same. There’s inclusion each way, even if intuitions differ. Calculating is calculating no
matter how you decide to do it.
Is this the final word? Probably not. Everyone seems happy with two cultures—
the visual and the verbal—and bent on keeping them so, separate and equal. But it
does go a long way toward showing that this is pointless and ironically reductive—
calculating is just moving symbols around. Yes, it is—and look at what you can see and
what you can do.
I Don’t Like Rules—They’re Too Rigid
The schemas I’ve used so far define rules of very broad, overlapping kinds. There are
the identities
x fi x
and, more inclusively, transformations of shapes
x fi tðxÞ
Then I have schemas to define rules for myriad patterns with symmetry
x fi x þ tðxÞ
for fractals
x fi X tðxÞ
and for adding
x fi A þ B
and subtracting
A þ B fi x
according to spatial relations. And, finally, there are rules connecting shapes and the
shapes they bound—for example
x fi bðxÞ
to show the boundary of x, and
bðxÞ fi x þ bðxÞ
to show bðxÞ and a shape it bounds.
All of these schemas are worth having. Nonetheless, they’re too loosely drawn to
be decisive in practice to get shapes with specific properties or, more ambitiously,
designs. My schemas give vague hints, not definite solutions. I can be perfectly general
about this with the schema