Shape

55 Background
and time. The way Euler came to describe polyhedra in terms of their vertices, edges,
and faces—in terms of boundaries, boundaries of boundaries, etc.—is also enlightening.
33 Peter Cromwell goes on to describe the tactile quality of these elements in the
short quotation on page 32. 34 And Erwin Panofsky mentions Albrecht Dürer’s
‘‘nets.’’ 35
Euler and William Blake have the same politics. And Blake is a wonderful letter
writer, too. His displeasure with ‘‘Newton’s Doctrine of Fluxions of an Atom’’ and his
stirring vision of the ‘‘Republican’’ line were conveyed on 12 April 1827 to George
Cumberland. 36
The quotation from C. S. Peirce and the twin figures
are from his famous article in Popular Science Monthly. It’s called ‘‘How to Make Our
Ideas Clear.’’ 37 Ambiguity and clarity are opposites—it’s one or the other. One way to
be clear is to see that figure 1 and figure 2 show the same arrangement of objects. Only
how can I tell I’m looking at an arrangement and not at a shape? Maybe if I divide a
shape into parts, it’s an arrangement of objects. But the shape comes first, and it’s there
to divide anew when I look again. Evidently, arrangements like sets are ideas and can
be clear, whereas shapes are ambiguous. What can be seen can’t be clear (unambiguous),
and what’s clear can’t be seen (ambiguous). Shapes aren’t ideas. Was Plato right
that philosophers make better rulers than ‘‘those who wander in the region of the
many and variable’’? Is seeing mindless? Thinking with your eyes and dealing with
ideas aren’t the same. And sure enough, Peirce shows why in his neat discussion of
habit, after the passage I’ve cited. It’s another way to describe rules for calculating.
What the habit is depends on when and how it causes us to act. As for the when, every stimulus to
action is derived from perception; as for the how, every purpose of action is to produce some sensible
result. 38
The arrangements in Peirce’s figures are identical. But if perception is the test, then
each supports different behavior. Is this a mistake? The philosopher finds ambiguity
embarrassing and shuns it in logic and reasoning, even as the shape grammarist
embraces ambiguity and exploits it in art and design. Rules are used in an open-ended
process to decide when and how to change shapes with all of their ambiguity. It’s asking
too much to ask for less, and this is true when less is more. Doesn’t anything hold
still? One thing is clear. Arrangements like definitions and plans are mistakes—all four
are made to check creative activity.