Graphic representation and graphic design

If, as its effect on the development of linguistic theory indicates, graphic representation furnishes the models for linguistic science, we must now ask how it does so. How does a diagram serve as the analogue for a linguistic explicandum?How does it provide the neutral analogy whose exploration leads to the modification of linguistic theory? To answer such questions we must inquire into the nature of graphic representation, taking the second of our two perspectives, the principles of graphic design.

Graphic representation and the principles of graphic design

Graphic representation is a language, and the principles of graphic design are its grammar. Because figures are made to stand for something else, however, the relationship between form and meaning is more complicated than it is for natural Ianguages. Graphic representation is in fact more like a code than like natural language; for it bears at once its own meaning--its meaning as an utterance in a language--and the meaning grafted on it by its use as a symbol--its meaning as an utterance in a code. We shall call the two kinds of meaning "visual meaning" and "symbolic meaning." But there is still another aspect of the meaning of a figure, what might be called its connotation, as opposed to the first two kinds of meaning, which are its denotation. For just as an utterance in a natural language carries expressive overtones not governed by its denotation--harmonics, as it were, of the note sounded by its denotation--so a figure carries expressive value. This we shall call "expressive meaning."

Visual meaning

Like any language, graphic representation has a vocabulary and a grammar. The vocabulary of graphic representation is the "art elements."¹⁷ Some of them must be ruled out for linguistics on grounds of impracticability; thus, from the complete inventory of art elements,

The last two, volume and movement, must be deleted: volume, because graphic representation is not plastic; movement, because it is not dynamic. Three more are ruled out on grounds of impracticality: texture, hue, and intensity are difficult and expensive to reproduce. The four remaining art elements--line, shape, space, value--are the vocabulary of graphic representation for linguistics. Though they are not wholly independent of each other, they enjoy a limited autonomy. There is line, an independent element in which value (relative lightness or darkness) may or may not be significant. There is shape, in which value may or may not be significant, but which surely depends on line for its realization. And there is space--two-dimensional for our purposes-- against which the presence of line and shape contrasts with their absence.

Klee sees the relationship between line and shape as more complex. He distinguishes three "characters" (1961:115):

"linear character" (linear-active, planar-passive):

Configurations of a middle character are the ones we shall have most to do with. For these, Klee gives three "basic forms," each of which may be construed in two ways (32):

There are, then, things a particular shape can be made to say,and things it will say, despite the designer's intentions to the contrary.

The four art elements are manipulated according to the grammar of graphic representation, the principles of composition. Different treatments of the subject list different principles.¹⁸ There are three, however, that are indispensable: balance, rhythm, and emphasis.

Balance is of two sorts, overt and "occult" (Scott 1951:43-46). Overt or "static" balance (Kepes 1944:36) comprises "axial balance" and "radial balance" (Scott, 43-45). Axial balance corresponds to Weyl?s (1952) bilateral symmetry; radial balance is his cyclic symmetry, the symmetry of snowflakes and rose windows. In overt balance, like things balance each other along axes or around a focal point; occult or "dynamic" balance, in contrast, is the product of elements that are, not like, but unlike--"a pound of iron balan- cing a pound of feathers" (Kepes, 36). The numerous ways in which occult balance is achieved all involve compensation. The art elements compensate one for another: a small dark shape, for instance, balances a large light one. It is easy to see that occult balance has tension, dynamism, a sort of centrifugal force; it will suit some meanings and not others. We can sum up the various kinds of balance as

Rhythm--the second design principle--is, in a word, repetition: it is " expected recurrence" (Scott, 63). It corresponds to Weyl's translatory symmetry, the symmetry of frieze designs,

(Weyl 1952:49)

and centipedes:

(Weyl 1952:5)

There are several kinds of rhythm (Scott, 63-65) besides repetition of the same unit or interval: progression of identical shapes of increasing or decreasing size; alternation of dissimilar units; and occult rhythm, the repetition of "whole systerns of relationships." Simple repetition creates what Klee (1961:230-231) calls "dividual" structures, like the cellular structure of a flower. Progression, alternation, and occult rhythm create ״individual" structures, like the flower itself. Addition or subtraction of a unit leaves a dividual structure unchanged, but turns an individual structure into a new form (Klee, 257). We can sum up the various kinds of rhythm as

The third design principle is emphasis. The means by which emphasis, overt or occult, is achieved are infinitely various; but always, as balance depends on compensation and rhythm depends on repetition, emphasis depends on contrast. Part of a design may be darker (or lighter) than the rest; or larger (or smaller); or its position may give it emphasis. The focal point of a design with overt balance is naturally the center, just as the last word in a sentence naturally receives emphasis. The focal point of a design with occult balance is created by the arrangement of elements within that design, as the first word of a sentence is given emphasis by inverting word order.

The vocabulary of graphic representation, then, comprises the art elements of line, shape, space, and value. The grammar of graphic representation--the arrangement of these elements--follows the design principles: balance, rhythm, and emphasis. But we have not accounted for all the ways in which a figure makes a meaningful statement. So far we have been looking at part of the meaning of graphic representation, what we have called visual meaning. In reality, of course, visual meaning is inextricable from expressive and symbolic meaning. Nevertheless, the existence of visual meaning is demonstrated by the possibility of visual ambiguity. The right-hand figure in each of the following pairs is visually ambiguous :

(Arnheim 1964:9-11)

We cannot read the right-hand figures with any certainty: are they meant to have the assymmetrical proportions of the figures on the left in each pair?--or are they intended as symmetrical:

By their visual meaning, figures make a statement that we can weigh, doubt, argue about, without appealing to anything outside the figures themselves: we do not need to ask what they stand for.

Expressive meaning

Expressive meaning resides in the "expressive value" (Watkins 1946:10) of the art elements out of which the message is constructed, and the connotations accruing from the culture in which sender and receiver participate. The expressive value of line (Watkins, 10) is:

The symbols get their expressive value from their function as a "simplification, or graphic shorthand" for the natural world (11). Cultural connotations are likely to echo expressive value: connotations of eternity adhere to the circle from the mandala in Oriental religion; other connotations adhere to the tree in western culture from the Tree of Knowledge and the Cross.

Expressive and cultural associations are communicated by a design independently of its subject matter (Anderson, 60). The expressive mean- ing of the circle, for instance,

(Schleicher 1888:97)

is eminently suited to an eternally repeated process. It is the expressive meaning of a design, moreover, that furnishes metaphor; if the expressive meaning takes over the figure, unwarranted development of theory can result, as with the tree in synchronic linguistics.

Symbolic meaning

The symbolic meaning of a design is its meaning, its effectiveness as a symbol. A figure must reconcile its intended meaning, its subject matter, with the visual and expressive meaning of its design; their congruence is the measure of a good design. It is a question of translation. Like a good translation, a good representation is isomorphic with its subject matter: in it, as Y. R. Chao puts it, "symbol complexes [are] in iconic relations with object complexes" (1968:221).¹⁹ We shall call such a figure an isomorph. The idea is not new. Bell (1867:35) gives it as "the fundamental principle of Visible Speech that all Relations of Sound are symbolized by Relations of Form." We have arrived again, by a different route, at the question of whether graphic representation occupies in linguistic theory the place of models in scientific theory. For what is the conception of the model as a material analogue but a statement of the principle of iconic relations? Indeed, Chao' s (1962) study of the senses of the term "model" indicates that the one element common to them all is "some structural similarity shared by two things" (1968:202).

An isomorph, Chao takes pains to point out, is to be distinguished from an icon. The former is the "relevance of the structure of symbol complexes to the structure of objects"; the latter, the "fortuitous relevance between a simple symbol and its object" (1968:220). Many road signs are icons. The symbols > and <, in both mathematics and linguistics, are icons. The Chinese box diagram is an icons. in that it "clearly indicates.the notion of constructions being nested" (Gleason 1965:158). It is at the same time an isomorph insofar as it represents similar relations by similar means, so that it recapitulates the structure of the explicandum.

Chao dismisses iconic symbols as essentially unimportant, and no doubt they are a luxury; nevertheless, for a theory of graphic representation in linguistics, iconicity is as important as isomorphism. Its importance is negative. Isomorphism characterizes a figure that will be fruitful for theory--that furnishes a neutral analogy at least some aspects of which will ultimately be part of the positive analogy. Conversely, negative analogy is to be shunned. Thus what Chao (1968:219) calls "irrelevant iconic features" loom large. As taking the metaphor literally is a sort of imperialism of expressive meaning, so taking the figure literally is a sort of imperialism of visual meaning. Visual meaning, the figure as a design, usurps the place of symbolic meaning; iconicity supplants isomorphism. The result, as we saw for the tree, can be the unwarranted development of theory.

Our theory of graphic representation for linguistics, then, comprises the aspects of graphic design set out in the chart on the following page.

The design of graphic representation for linguistics

Now let us see what our theory of graphic representation can do to unravel the tangle of correspondences between form and meaning that exists for graphic representation in linguistics:

What we would like to do is remove the synonymy and homonymy-- the knots, as it were, that occur at the points marked "tree," "componential analysis," and "block diagram." We shall not take the Gordian way out, but rather use our theory of graphic representation as a principled basis for the unraveling.

At the same time, we shall be looking at graphic representation from the other of our two perspectives, as a component of linguistic seience. Indeed, we cannot avoid doing so: for the aspect of the meaning of a figure that we have called symbolic meaning--which presumes the relationship of material analogy between the figure and something else--is the very aspect of its meaning by virtue of which we can speak of graphic design at all. "Design" implies a match between figure and meaning, as "scientific model" implies a match between analogue and explicandum.

The tree design

Line and space are the art elements from which the tree design is built. Its balance, insofar as the principle of binarity is followed, is bilateral symmetry; its emphasis is the unique beginner. But the most interesting thing about the design is its rhythm. The tree has what Klee calls individual rhythm. This is true even though it clearly has a sort of translatory symmetry--that is, it replicates itself along the vertical axis :

Because a tree has a top and bottom, and often a left and right as well, adding to it or subtracting from it changes it. Even rearranging it makes the figure take on a different shape:

The tree is therefore peculiarly suited to representation of things haying an inherent order or hierarchy of some sort. The repetition of units--of the combination of node and branch -- is more than simple addition; in the tree, repetition is recursion. Such stubbornly asymmetrical (in the mathematical sense) notions as genesis, taxonomy, constituent analysis--all are reflected in the individual rhythm of the tree; for one cannot subtract from, add to, or rearrange the units of such systems without toppling the whole structure. The tree design captures the intransigence of time and space.

The tree is as unsuited to componential analysis as it is suited to the other three meanings. A key--a tree for componential analysis-- is characterized by the interchangeability of rows; but we have seen that rearrangement of units is counter to individual rhythm. A key is characterized by overlapping classes; but we have seen that a symmetrical relationship among units is counter to individual rhythm. A key is characterized by items all of equal rank (all nodes are features, all terminal elements are sounds); but we have seen that equality of units is counter to individual rhythm. The characteristics of a key are in fact just the characteristics of dividual rhythm. A dividual structure, then, is the kind of design we should look for to express componential analysis, and not the tree at all.

Eliminating componential analysis from the list of meanings for the tree, however, still leaves us with homonymy. We have already discussed the problems presented by the tree design for the other three meanings. For taxonomy, there is the problem of overlapping classes: in linguistic taxonomies, some overlap seems to be inevitable. But this we can solve by the use of "Rorschach" diagrams like the one of Pike's reproduced in Chapter 2 and the ones for symmetry and rhythm given in this chapter. For genesis, however, there is the problem of the visual meaning of the tree; as we have said, the limitations of the comparative method--abrupt and simultaneous separation of daughter languages from the parent and from each other--are largely the limitations of the figure. For constituent analysis, there is the problem not only of the influence of the figure on the development of grammatical theory (such notions as extraposition, ordered elements in the deep structure, and so on), but also of the influence of its metaphor (such notions as pruning, chopping and grafting, and so on); that is, both the visual meaning and the expressive meaning of the tree design have furnished a neutral analogy that ended in the unwarranted development of linguistic theory.

One proposal for circumventing the limitations of the tree for historical linguistics is Southworth's amalgam of tree and wave diagrams (Southworth 1964; quoted in Anttila 1972:308-309). A wave diagram allows the representation of overlapping relationships -- the point of Schmidtf's Wellentheorie--not expressed in a tree:

(Schrader 1883:99)

But it cannot replace the tree. The tree, in Southworthfs words (558) , ״shows splitting processes but cannot show overlapping relationships, whereas an isogloss map shows all the relationships at one particular stage of the history, but cannot show more than one stage." SouthworthTs object is a new design that gives na truer and clearer picture of linguistic history" (557). The amalgam (actually its penultimate version, but the best one) looks like this:

(Southworth 1964:562)

The wave component, in rectangles, expresses time only indirectly, by length along the vertical axis of the tree: the longer the rectangle, the earlier the phenomenon occurred. The wave component largely takes over representation of Nebeneinander; the tree component represents Nacheinander.

If the construction is unwieldy enough to limit its usefulness and to prevent its substitution for tree diagrams with the genetic meaning, it is not the underlying theory that makes it so. On the contrary, it is legitimate, and certainly not unprecedented, to combine different models for a single explicandum provided they are complementary--the wave and particle models in physics, for instance (Hesse 1966:53; Toulmin 1953:35). Now, Schleicher's and Schmidt's versions of linguistic history, as Leskien first pointed out (Anttila 1972:307), are complementary. It is reasonable to assume that the two figures are also complementary. Perhaps the problem is one of design? "Bilder," Schmidt himself said (1872:28), "haben in der Wissenschaft nur ser geringen wert." Indeed, it is hard to imagine Southworth's figure represent- ing, say, all the isoglosses for Indo-European. It is harder still to imagine it showing the particularity of Hockettfs "detail" from a wave diagram for the Germanic languages,

(Hockett 1958:69)

Let us see what our theory of graphic representation has to say about it. What are the art elements of which the design is constructed? No sooner is the question asked, than we have found the problem: a single element is employed in addition to space, and that is line. The rectangular shapes are simply line configurations, forms of what Klee calls a middle character; the nodes are simply points on a line. Clearly this is overburdening a single graphic device. We have met this problem before, and shall meet it again. Most of the figures used in linguistics, all of which are two-dimensional, are constructed from line and space. The vocabulary of graphic representation was limited at the outset to four of the art elements: line, space, shape, value. Now we find it has shrunk to two. Though our theory has failed to provide an answer to these difficulties, it has provided the questions.

Visual meaning is as troublesome for 1С representation in general as it is for tree diagrams with the meaning of constituent analysis. Attempts to better the match between figure and explieandum like those that follow all address themselves to the same problem and use the same means.

(Nida 1966:21)

(Hockett 1958:247)

(Lamb 1966:25)

The diagrams of Hockett and Nida introduce additional graphic devices for endocentricity and exocentricity, discontinuous constituents, and parataxis. The system of representation devised by Lamb for stratificational grammar (like that of Hudson [1971] for "systemic" or neo-Firthian grammar) solves the problem of discontinuous constituents with graphic devices allowing the branches from a node to be either disjunctively or conjunctively ordered and either simultaneous or sequential. In addition, the greatly modified trees of stratificational grammar--when they represent not simply a single sentence, like the diagram above, but a section of the grammar-- incorporate other changes. They can be bidirectional, somewhat in the way of the double tree from Pike reproduced in Chapter 1; and they can have more than one beginner, somewhat in the way of Morin and O'Malley's vines, at points where the Mknot pattern" combines input from two sources, the "alternation" and "tactic" patterns:

(Lamb 1966:17)

Essentially, the problems dealt with by the modifications of Nida, Hockett, and Lamb are all different faces of the same thing. This is at the root, as well, of the difficulties presented by the influence of the tree figure on the development of transformational-generative grammar. It is the linearity of graphic representation. That it is a defect of graphic representation in general rather than the tree diagram in particular is supported by the fact that other kinds of 1С representation contrive devices to circumvent the same limitations; for instance, Francis modifies the Chinese box to represent endocen- tricity and exocentricity:

(Francis 1958:313)

Linearity not only imposes an inescapable order on constituents, but also entails continuity and hypotaxis. For in a chain, contiguous elements are necessarily connected, and noncontiguous elements necessarily unconnected; and only the introduction of special symbols can, by an act of representation, as it were, abrogate these relationships. Adjacency or enclosure within the same block or box cannot mean at once hypotaxis and parataxis, endocentricity and exocentricity, membership in the same construction and membership (because of a discontinuous construction intervening) in two different constructions. Special symbols are required as long as representation is two-dimensional.

Expressive meaning also presents difficulties for the tree. As we have seen, it furnishes the diagram with a metaphor which at times usurps the place of the analogue.

Commending Reed-Kellogg diagrams is rather like commending the English writing system, orthography's ugly duckling; but for the representation of constituent analysis, old-fashioned Reed-Kellogg diagrams might well be the best we can do. For one thing, the Reed-Kellogg system is more commodious than any we have looked at. As Gleason (1965:142-151) points out, it covers everything from

(Gleason 1965:151)

If line and space are the only art elements pressed into service for graphic representation in linguistics, Reed-Kellogg notation at least exploits them to the fullest. More than that, Reed-Kellogg diagrams have several things to recommend them. First, the figure is neither a tree nor a block diagram (though it has elements of both), and so does not participate in homonymy. Second, it represents simultaneously both surface structure and deep structure, something none of the other figures we have considered does: any constituent perched on a sort of two-legged stand is an embedded sentence (though the converse does not hold). The other 1С representations show either surface structure or deep structure, but not both at the same time. Finally, and perhaps most important, Reed-Kellogg diagrams support no metaphor, neither fauna nor flora.

It is true that Reed-Kellogg diagrams, like the others, are cumbersome, heavy with the weight of auxiliary notation, special symbols and the like; for they too cannot represent discontinuity; they too are subject to the constraint of linearity built into graphic representation. Like Southworthfs proposed modification of the tree for historical linguistics, Reed-Kellogg diagrams lose in simplicity what they gain in accuracy. We have not succeeded in finding a replacement for tree diagrams for either the genetic meaning or the meaning of constituent analysis; and it is beginning to look as if the exigencies of a two- dimensional medium are insurmountable.

The matrix design

A design very like the matrix,

(Klee 1961:217)

is characterized by Klee (1961:217) as '1the most primitive structural rhythm"; it is, he says, no more than "a combination of the two directions," "an addition of units in two dimensions." It is in many ways the direct opposite of the tree design.

The meaning of the matrix figure lies in its dividual rhythm. Klee might have said, the relentless addition of units; for this, as we have seen, is what makes the figure. It gives it its characteristic balance, its translatory symmetry; the matrix is translatory in both directions, horizontally and vertically, outdoing in this respect even the centipede, which is translatory in one direction only. The absolute likeness of each cell to every other comes from translatory symmetry--and a fearful symmetry it is, since it appears to determine the figure*s great negotiability. The matrix is expandable; it is modular; it is the blueprint for its own expansion. But so is the tree; and we have noted the coincidence of warranted extension with the matrix and unwarranted extension with the tree. How do the designs differ? Repetition of units in the matrix is the simplest sort of propagation, addition. Repetition of units in the tree is recursion, a complicated sort of propagation which locates each successive repetition within an unfolding hierarchic structure. Repeating a system of relationships (the configuration of node and branch), like building a house of cards, precludes a democratic interchangeability of units. Conversely, repeating identical units (cells) leaves no room for the expression of hierarchy.

So it is that the tree design with its individual rhythm is not suited to the representation of interchangeable units, but the matrix design with its dividual rhythm is made for it. We have called the cells of the matrix the product or intersection of the two dimensions; we might instead have called them the overlapping of two classes, since every cell is a point at which two criteria overlap. In this the matrix is diametrically opposed to the tree; for not only taxonomic trees, but trees of any sort, are ill-formed if they represent the intersection of classes. Therefore the notion,

is better put as

--not as a tree but as a matrix (Kay 1969:81; Pike 1962:230-231; Wallace and Atkins 1960:409). Representing it by a tree entails the extension of the theme of constituent analysis to a concept that falls within the other great theme of componential analysis: it entails the collision of two opposite theories.

The dividual rhythm of the matrix design gives it its characteristic emphasis as well. What fills the cells is the marrow of the figure. It is for this reason that the grid deploys the elements of componential analysis less than strategically. In the grid, the arrangement of elements--sounds, features, feature specifications--is at odds with the figure's emphasis. What fills the cells is a collection of uninterpretable symbols (+, -, 0)-- uninterpretable until decoded by recourse to the elements on the perimeter.

The extreme simplicity of the matrix design leaves it without expressive meaning. It is, after all, no more than the juxtaposition of the two dimensions of the pictorial surface--the bare bones of graphic representation. Yet it is perhaps its very inexpressiveness that has endowed it with a heuristic usefulness for linguistic theory. Its static quality, for instance--for unlike the tree, it has no built-in directionality--is what suggested the theories of Pfalz, Austin, and Martinet, the notion of sound change as movement. Certainly its simplicity facilitated its extension--and the concomitant extension of componential analysis-- to different kinds of linguistic explieanda. At any rate, it has not proved the fertile ground for the growth of metaphor that the tree has been.

The matrix design gives iconic features aí֊ most as little play. Putting outside the matrix units that stand outside the system (as in the figure from Trubetzkoy) uses the matrix as an icon; so does letting the order of columns follow the oral cavity, so that left to right reflects front to back. And matrix arrangements of vowels tend to turn into schematic representations of the oral cavity. Thus

can transform itself into

with chameleon rapidity. On the whole, however, symbolic meaning does not present problems for the matrix design.

The box design

The cube and the matrix are synonyms; but they are also complementary. The cube accommodates three features; the matrix, two. Thus we shall not be concerned to eliminate synonymy, but, if we can, to exploit it.

Potentially misleading elements of visual meaning for the cube are its rhythm and its emphasis. The matrix has dividual rhythm; the cube has individual rhythm. It exhibits occult repetition, as the trouble we had resolving it into six occurrences of a single shape testifies. We could prize it apart only by using a special tool, the concept of projection to different planes, thus setting the figure apart from all the others. The difference between the cube and other figures makes it hard to see the cube as the logical extension of the matrix, as simply a more capacious matrix. This becomes clearer when we consider extending it one feature further. Where can the figure go? We can of course resort to graphic representation of four-dimensional constructions, like this one:

(Lotz 1967:3)

But representation of a four-dimensional figure in a two-dimensional medium makes a configuration almost impossible to read; and this is compounded by the fact that, like all the other figures, it is built out of only two art elements, line and space. Moreover, where is the figure to go from here? What about five features? Six?

The problem of the figure1s emphasis is part of the larger problem of its rhythm. Though componential analysis is a dividual concept, the cube is an individual structure. The front of the cube is the focal point, because it is the only side seen clearly and non-obliquely (the back, not seen obliquely, is nevertheless obscured). Whatever is placed at the front qf the figure receives emphasis; thus, though the ciibe does not, like the tree, impose a hierarchical arrangement on the data, it does throw part of the data into relief-- generally undesirable for thq expression of componential analysis.

Perhaps the most interesting aspect of the difficulties presented by the visual meaning of the cube is its ambiguity. Because only the two art elements, line and space, construct the figure, the cube, like the examples of pure design we looked at earlier, is visually ambiguous. The cube may easily dissolve into a collection of points joined by lines, as a slight distortion makes clear :

The figure wavers between two־ and three-dimensionality: it may be read as a seine-like structure, when it is intended as a structure of slab-like pieces. The consequence for its symbolic meaning is that the componential analysis is carried by the lines between the points (properly speaking, the corners); whereas actually, as we noted, the sides of the construction carry the analysis. That cube diagrams are nonetheless intended to be read differently from network diagrams is clear from a comparison of the two. Jakobson*s cube diagram for the componential analysis of Russian declension, for instance,

(Jakobson 1963:149 [Roman letters replace Cyrillic])

looks very different from a network diagram for the same thing:

This network version of the same componential analysis rests on point and line: points represent the units of the system, the cases; lines represent shared features.

Value used together with line transforms the cube into an unambiguously three-dimensional configuration,

but destroys its transparency and with it the armature of three-feature analysis. Forchhammer's early version of the cube, probably the prototype, uses line rather than value to effect unambiguous three־dimensionality :

This preserves its transparency, but leaves the same problem as did the modifications proposed for 1С representation--too great a burden on the single art element of line.

The expressive meaning of the cube poses no problem; nor does it, like the tree, carry a metaphor. As for its symbolic meaning, the iconicity of the cube is often squandered. In the cube diagrams of Hockett and Gleason, reproduced in Chapter 3, what a chance is missed by not putting the front vowels at the front of the figure, the back vowels at the back!

The second sort of box diagram, the Chinese box, has static balance and radial symmetry, like Klee's figure,

It consists in the repetition of a single shape of what Klee calls a middle character. With the treelike scaffolding of all 1С representation, the Chinese box, like other figures for the meaning of constituent analysis, is an individual structure. This is because its overt repetition is not simple repetition, but progression: the repeated squares are graduated in size, so that as one reads out from the center of the figure they get larger and less specific. The concentricity of the repeated shapes creates the figure!s emphasis. Its focal point is the center of concentric rectangles, the core of the figure.

The expressive meaning of the Chinese box includes its likeness to Venn diagrams, a tool in the algebra of sets and symbolic logic for the expression of inclusion relations:

Both express the implication: love →vegetable →шу; reverse this, and bending the arrows of implication a little gives an endocentric con- struction:

Here, then, expressive meaning reinforces visual meaning, and the concentricity of the figure is thus a formidable obstacle.

As for its symbolic meaning, the concentricity of repeated rectangles, which creates the figure's emphasis, makes expression of endocentricity (Hockett 1958:189) and exocentricity inherent in the Chinese box. A construction's endocentricity is automatically entailed by the diagram,

just as constituent function ([NP,S] [NP,VP] and so on) is automatically entailed by tree diagrams This has its drawbacks: even so simple a construction as

is at odds with the insistent singular emphasis of the figure.

Finally, let us look at the block diagram, the last of our three kinds of box diagrams Though it can be isomorphic with either a tree or a matrix, the block diagram is better suited to the meaning of conflation than to that of constituent analysis: its symbolic meaning is a conflated matrix by virtue of the fact that its visual meaning is a conflated matrix. Its visual meaning is a conflated matrix through converting a dividual structure into an individual structure. The underlying matrix expressing a full system has dividual, overt rhythm; conflation, erasing some of the boundaries between cells, converts this to individual, occult rhythm--a rhythm literally obscured. Clearly, the block diagram for the meaning of conflation serves Chao1 s principle of iconic relations. The figure preserves the matrix in all respects save those which have altered with time or from system to system, and whose altera- tion is the subject of the figure's statement.

We have unsnarled the correspondences between form and meaning to some extent:

We have done so by means of several deletions: the grid, because it is an ineffective use of the matrix design; the key--the tree for componential analysis--because the tree design is an irremediably bad fit for this meaning; the cube, because it is an individual structure used for a dividual concept; the Chinese box, because the neutral analogy it furnishes is likely to be misleading; the block diagram for constituent analysis, because the figure is a better fit for the meaning of conflation. We have made only one addition, the double tree for taxonomy.

This revision is unsatisfactory. For one thing, the tree is still a homonym: neither Southworth's proposed figure for the genetic meaning nor Reed-Kellogg notation for the meaning of constituent analysis is a good replacement for the tree. For another, the problems we noted for the neutral analogy presented by both the tree and the matrix remain unresolved. And finally, the exigencies of graphic representation--two-dimensionality and the scant vocabulary of art elements--are still a problem. In fact, what is needed is not fewer figures, but more.