Shape

44 Introduction: Tell Me All About It
she replied, ‘‘three.’’ I liked this—a desk is a table, isn’t it?—but Catherine’s teacher
wasn’t pleased. I thought it might be the room—it was a bigger rectangle. Or was the
small rectangle on her teacher’s desk a table? Maybe the room and the tables were
desks—how many did her teacher have? Somehow, I missed the point. I was looking
at it the wrong way. Flags, pencils, and globes weren’t wastebaskets. And pencils—
long, thin things—couldn’t be globes—round, fat ones. This was right. Only Catherine
didn’t care. She looked at the six (?) drawings in the key and saw shapes. No one told
her not to. She used her eyes to count. Rectangles made tables and desks. I didn’t ask
about things with circles. The room was magic—embedding was involved. It changed
every time she looked. What good were standards now? But counting required identity.
Why was this so important? Who said maps and plans worked that way? Were they
hundred-squares? Why not the other way around, so that grids were to see? What happened
to ambiguity and reasoning? (A math specialist at Catherine’s school tells me
she’s amazed at how many children say her room has changed whenever they visit
again. This may complicate counting or cause problems in metaphysics—it shows
how artificial they are—but it doesn’t worry children. The rooms they see are the
rooms they draw.)
Alexandra and Catherine don’t make definite things. They draw lines to see what
they get—Catherine actually says so—and blithely go on from there to try it again. It’s
like an experiment, or more like painting. The outcome can’t be known beforehand because
there isn’t time to look. Within the lines, Alexandra and Catherine are free to see
as they please. It’s easy to find new things in new situations. They’re dealing with novelty
all the time. Is this something to use? Seeing and counting aren’t the same, but
there’s reasoning—yes, calculating—in both. Why not take advantage of it? To insist
on counting may limit children, if they learn with their eyes and on their hands (digits).
And it isn’t necessary when you can go on from what they see. Seeing is as useful
as counting. It’s thinking, and there’s a lot to show.
It Always Pays to Look Again
This book is divided into three parts that correspond pretty closely to the three stages
in my autobiography. But the parts are presented in reverse order. I need to show you
more about what it means to calculate with your eyes—how seeing and counting differ
when you use rules—before I can show how to calculate with shapes. Then there are
the formal details to make sure that it works. And you need to know how to calculate
with shapes before you try it out to design. So the three parts are these—
Part I:
Part II:
Part III:
What Makes It Visual?
Seeing How It Works
Using It to Design
‘‘It’’ is in the titles on purpose, and sometimes it and pronouns like it are elsewhere and
have no set antecedent. It can’t be helped with shapes and rules—what you see is what
you get, whatever was there before—and it can happen with words. Ambiguity is part
of everything I do, so I try to use it.