Shape

78 I What Makes It Visual?
formula to get to the idea that calculating is visual if it can deal with shapes like this. I
want to use rules to determine what parts I see and what I can say about them, and to
allow for what I see and what I say to change freely as I calculate. And I want this to
happen every time I try a rule. I’m told calculating is a good example of what it means
to be discursive—well, sometimes when I calculate it looks logical, but most of the
time it’s only desultory rambling with my eyes. Shapes should be ambiguous and
vague—full of miscellaneous possibilities—and ready to use when I calculate, wandering
in the region of the many and variable.
So what can I do to make Evans’s example visual? My formula provides twin
options. I can change the elements in Evans’s shape from lines to points, so that
dimðelÞ ¼dimðemÞ, or I can use another embedding relation that’s one dimensional.
Each of these alternatives is feasible and amply rewards a closer look.
Suppose that the shape
is the nine points Evans uses to define line segments as constituents
and that the embedding relation is the same—I’ll continue to require identity among
constituents. But with points, I really don’t have much of a choice. Embedding works
only in this way.
I can use the rule
Three points fi Triangle
in place of Evans’s rule to define 45-degree right triangles
Or equivalently, I can give the rule in this identity
in the way I normally do in shape grammars, where two shapes—in this case, they’re
the same—are separated by an arrow. I’ll say more about rules like this a little later