Shape

371 Getting in the Right Frame of Mind
while the schema x fi x and its inverse are equivalent, along with the twin schemas
x fi tðxÞ and x fi xA and their inverses. But notice, too, that in practice bðxÞ fi x works
for x þ bðxÞ fi x—here, for a square and its boundary
x fi divðxÞ
I’ve also tried a few general-purpose schemas of another sort. For example, there’s
to divide shapes in various ways—typically, the shapes are polygons as in ice-ray
window grilles, housing plans, and paintings, although polygons aren’t required for
the schema to work—and then there’s my old standby from ‘‘Kindergarten Grammars’’
x fi A þ B
for spatial relations that are defined either from a given vocabulary of shapes, maybe a
square and a triangle
combined side to side in the following ten ways, where squares and triangles vary in
size and orientation
or from parts selected using identities in the schema x fi x, perhaps when x is a polygon
and identities are applied to the shape