Shape

394 Notes to pp. 50–51
same—try polygons, but also letters or words or Chinese ideograms, or other familiar things. Part
ambiguity is ‘‘special and consequential’’ in Knight’s synoptic taxonomy, and representational
ambiguity is ‘‘pervasive.’’ Yet my dialogue proves there’s dependence, too—the pervasive is a
consequence of the special. Klee draws lines to get planes, and surely he relies on the lines he
draws and the shapes he sees to decide what he’s shown. That’s why lines are medial, even if this
works in reverse drawing planes to get lines (boundaries). I guess all basic elements—points, lines,
planes, and solids—are medial in one way, while lines and planes are medial in two—this despite
the evident sequence in Wassily Kandinsky’s Point and Line to Plane. (At the time, the problem in
logic was to go from solids to points. Alfred North Whitehead’s method of extensive abstraction
did the trick, but for an inclusive review see A. Tarski, ‘‘Foundations of the Geometry of Solids,’’ in
Logic, Semantics, Metamathematics, trans. J. H. Woodger [Indianapolis, Ill.: Hackett, 1983], 24–29.
This is all pretty neat, only it isn’t about shapes. Basic elements of different kinds combine finitely
and not as members of sets. They’re independent and equal, and aren’t defined one kind from another.)
Then perhaps there are more types of ambiguity, for example, functional ambiguity where
the question is about behavior or patterns of events. But part ambiguity may be implicated here,
too. Also on page 13, I said that the above square-and-diagonals with small triangles is a pyramid
in Chinese or Renaissance perspective, and with large triangles, a Necker-cube-like tetrahedron.
On the pyramid, see El Lissitzky, ‘‘A. and Pangeometry,’’ in El Lissitzky: Life, Letters, Texts, ed. S.
Lissitzky-Küppers (London: Thames and Hudson, 1968), 349.
13. ‘‘Transparent construction’’ was rife at the Bauhaus—in particular, ‘‘Kandinsky . . . was one of
the leaders of the constructivist curriculum . . . [and] shared the basic faith in a building up from
the elementary.’’ Galison, ‘‘Aufbau/Bauhaus,’’ 738.
14. G. D. Birkhoff, Aesthetic Measure (Cambridge, Mass.: Harvard University Press, 1933).
15. G. Stiny and J. Gips, Algorithmic Aesthetics (Berkeley: University of California Press, 1978). A
worked example of E Z is given in G. Stiny and J. Gips, ‘‘An Evaluation of Palladian Plans,’’ Environment
and Planning B 5 (1978): 199–206.
16. O. Neurath, ‘‘Visual Education: Humanisation Versus Popularisation,’’ in Encyclopedia and Utopia:
The Life and Work of Otto Neurath (1882–1945), ed. E. Nemeth and F. Stadler (Dordrecht:
Kluwer, 1996), 330. Galison, 709–752, discusses Neurath’s role in the Vienna Circle and his relationship
to the Bauhaus. Moreover, ‘‘Neurath’s pictures were intended as clear, universal building
blocks on which all else could be built. . . . [They were] essentially a linguistic [syntactic] and pictorial
form of transparent construction.’’ (Galison’s phrase ‘‘pictorial form of transparent construction’’
is ironic in the same way the term ‘‘shape grammar’’ is, but nonetheless it’s consistent with
Neurath’s idea of a picture—and Wittgenstein’s, for that matter. This isn’t the case for the title of
my book Pictorial and Formal Aspects of Shape and Shape Grammars, where ‘‘pictorial’’ and ‘‘formal’’
anticipate the scheme for visual and verbal expression I outline on pages 7 and 8, and develop in
part II. Aspects of a Theory of Syntax is one of Noam Chomsky’s famous titles. Yet, even so, shape
grammars were never syntax—it’s always been shapes to shapes rather than symbols to strings.
The distinction is explicit in my Aspects, 28. The difference is huge when shapes aren’t symbols,
that is to say, when predefined units aren’t distinguished in shapes to add up to whatever there
is.)
17. E. Pound, ABC of Reading (New York: New Directions, 1960), 21–22.
18. M. L. Minsky, Computation: Finite and Infinite Machines (Englewood Cliffs, N.J.: Prentice-Hall,
1968).