Shape

54 Introduction: Tell Me All About It
Langer also separates technique and syntax in her discussion of drawing and language.
This is surely another version of the distinction between schemas and rules that
contrasts a synoptic view of possibilities with mere combination. In architecture, for
example, Gottfried Semper’s Urhutte is an ample summary of buildings in the manner
of Goethe’s famous Urpflanze, while J. N. L. Durand’s system of architectural composition
is combinatorial à la Chomsky or Simon with a vocabulary of building elements,
etc. Only the distinction isn’t that clear-cut—there are usually combinatorial aspects to
a schema when it’s used. Goethe relies on transformations and permutations to vary
the organs of his Urpflanze. Moreover, a system of vocabulary and rules with a starting
‘‘axiom’’ provides a synoptic view of possibilities that unfolds in a recursive process.
The axioms of any formal theory do this, as well. Maybe schemas and rules are alternative
ways of describing the same thing where the emphasis is differently placed. They
may be distinct, yet they both show what’s going on. Perhaps it’s only the way you
look at it, with the chance to switch back and forth. I’m used to conflating schemas
and rules when I calculate with shapes. I can look at shapes as schemas animated by
rules, or as something formed as rules are tried—either way without recourse to units,
so that there’s nothing explicitly combinatorial. And this is really what counts. In both
ways, there’s recursion, but there’s no vocabulary. It’s not words and rules. The shape
and rules to rotate and translate squares are a good example of this. Is the shape a
schema, an axiom, one then another at different times, or something else when it’s
used to produce the nine shapes on page 30 and others like them? 30
Herbert Simon’s The Sciences of the Artificial is an old book for computer science,
but it’s still current today in science and design. 31 Things never change as quickly as
people like to think, unless of course they’re shapes. Big ideas that are in right now
are in Simon’s book from the beginning in sharp focus, for example, complex adaptive
systems. I’m keen on the metaphor that calculating is painting—it’s new in the second
edition and stays in the third—even if Simon may miss some of its possibilities. I’ll get
back to it in part I, and to his way of describing pictures with hierarchies and his appeal
to drawings to disambiguate sentences. The latter may be a version of Wittgenstein’s
picture theory of meaning. I’m not sure. But I am sure that there’s a lot in Simon that
bumps against shapes in really important ways.
Leonhard Euler’s wonderful letters to the Princess of Anhalt-Dessau, the niece of
Frederick the Great of Prussia, are popular accounts of various topics in physics and
philosophy. The subject of ten letters from 25 April to 30 May 1761 is extension and
the absurdity of dividing things with it into other things without it. The hortatory
quotation on page 32 is from the penultimate letter of the series. 32 Euler’s warning—
deny parts that can’t be divided—applies to all sorts of things from shapes to numbers