Guest Essayist: Paul Lake
Paul Lake continues with the second part of The Shape of Poetry. Paul is not only a critic and poet; he's also a novelist. His satirical novel The Immortals (Story Line Press, 1994) will be reviewed here shortly.
Fractal scaling, another element common to both natural forms and fractal geometry, is the tendency of a fractal shape to be self-similar at various scales. For instance, the shape of a coastline remains the same no matter what the distance of the observer, whether he's a passenger on a jet, a mountain climber, a beach stroller, or child with a magnifying glass. Biomorphic shapes such as trees and leaves also display self-similarity at different scales: it's found in the fractal branching of limbs and twigs and in the vascular branching of its leaves. Another thing that causes the symmetry of natural forms to be irregular is the fact that some incorporate several fractal scales in their designs. For instance, human lungs have one fractal dimension for the first twenty branches, then a different scale after that.
As we've seen, the shape of a living thing such as a leaf is produced when the instructions coded in its DNA encounter chance elements such as wind, sun, heat, cold, and disease. Using a similar combination of coded rules and random elements, a mathematician named Michael Barnsley has invented a way to create remarkably leaf-like structures through a process he calls "affine transformations." As Briggs and Peat describe it, the process begins by shrinking and skewing the picture of a full-size leaf into ". . . a smaller distorted version of the original. The affine idea is to find several of these smaller leaf transformations that can be overlapped into a collage that has the shape of the original full-size leaf." After shingling several of these small distorted versions of the leaf together, Barnsley starts at a point on a computer and, "using his affine transformations and iteration . . . generates a fractal attractor that looks like the original leaf." Using a similar method (Barnsley calls it "collage theorem"), he has also created various computer-generated shapes of living ferns.
The poet Gerard Manley Hopkins, another astute observer of nature, seems to have discovered "collage theorem" on his own more than a hundred years before Barnsley. In his Journal entry of July 19, 1866, Hopkins describes what sounds awfully like the "affine transformations" for an oak leaf when he writes: "I have now found the law of oak leaves. It is of platter-shaped stars altogether." Hopkins also seems aware of other shaping principles in nature such as self-similarity and fractal scaling. Writing in his Journal again (July 25, 1868), he compares the shapes of clouds to patterns seen in fungus: "From the summit the view on the Italian side was broken by endless ranges of part-vertical dancing cloud, the highest and furthest flaked and foiled like fungus and colored pink." Benoit Mandelbrot, the inventor of fractal geometry, in his rhapsodic description of the fractal shapes of nature, practically echoes Hopkins' poem "Pied Beauty" when he writes of the ". . . grainy, hydralike, in between, pimply, pocky, ramified, sea-weedy, strange, tangled, tortuous, wiggly, whispy, wrinkled" shapes that can now "be approached in rigorous and vigorous fashion" through his fractal geometry. If we substitute the assonantally chiming word fractal for dappled in the first line of Hopkins' poem ("Glory be to God for fractal things"), its celebration of all that's "fickle, freckled (who knows how?)" sounds, in turn, remarkably like Mandelbrot.
What we are beginning to see, I hope, is that the laws governing the evolution of living and other natural forms are the same laws that govern the creation of poetry. So let's summarize what these laws are and show how they apply.
First, symmetrical forms such as sonnets, villanelles, and ballad stanzas are not static "received forms"; they evolve, like plants, through a process of iteration and feedback. The regular meter of formal poems is not a dull mechanical ticking, like a clock's; it coalesces out of the rhythms of randomly jotted phrases through a process of "phase-locking"--a natural process that occurs, in the words of Briggs and Peat again, "when many individual oscillators shift from a state of collective chaos to beating together or resonating in harmony." As two examples, they describe the way the randomly flickering lights of fireflies become synchronous throughout a whole tree, and the way the menstrual cycles of women living in close proximity often phase-lock into a single, collective rhythm.
Ralph Waldo Emerson, no champion of either conformity or mechanical regularity, has also suggested that meter and poetic form are inextricably tangled in an organic unity, writing, ". . . it is not meters, but a meter-making argument that makes a poem,--a thought so passionate and alive that like the spirit of a plant or an animal it has an architecture of its own . . ." A thought echoed by Whitman, who writes, "The rhyme and uniformity of perfect poems show the free growth of metrical laws and bud from them as unerringly and loosely as lilacs or roses on a bush, and take shapes as compact as the shapes of chestnuts and oranges and melons and pears, and shed the perfume of impalpable form."
Regular meter is also a source of feedback, another element of organic form. Once the random phrases and rhythms in a poet's scattered fragments begin to phase-lock into a recognizable meter, every subsequent line must take a similar form. During the composition of a metrical poem, the poet is often forced to rearrange and substitute words till the meter feels right. And when the meter begins to feel too regular, he will allow some random variations to occur, or consciously add substitutions, to reshape a line.
Rhyme is another attribute of formal poetry that increases feedback by forcing the poet to listen to what she writes. Each time it sends a poet to her rhyming dictionary, it adds an element of randomness--the way weather influences the growth of a tree. To fit the new rhyme into her scheme, the poet often has to invent new metaphors and introduce new ideas, thus altering the course of the entire poem. Furthermore, once the rhyme-scheme becomes clear to the poet in the early lines, it affects the development and arrangement of every future line as the poem evolves toward its final form.
In addition, rhyme prevents a poem from being merely a linear succession of so-called "breath units" scored on a two-dimensional plane. By reaching back to earlier rhymes and forward to future rhymes, each new rhyme twists the entire poem back on itself in a self-interfering knot.
A formal poem is also holistic. One alteration in a line's meter or rhyme scheme affects the writing, interpretation, and affective value of every other line.
The self-similarity of fractal scaling is another element shared by formal poetry, natural forms, and fractal geometry. We can see it in the scaled, fractal branching of the human circulatory system and in the identical paisley patterns seen at different scales of a purely mathematical object like the Mandelbrot set. The same self-similarity and scaling occur in formal poetry. Frederick Turner has found what might be the most remarkable use of self-similarity in all of literature in Dante's Divine Comedy, in which, Turner writes, ". . . the three line stanza of its microcosm is echoed in the trinitarian theology of its middle-level organization and in the tripartite structure of the whole poem." Narrative poems such as the Divine Comedy and the Aeneid of Dante's master, Virgil, offer additional levels of fractal scaling as episode branches into episode, creating eddies and turbulence that both contribute to and resemble the larger narrative movements of the poem.
The reason natural forms such as trees and leaves rarely display the perfect symmetry of a nautilus shell or the Mandelbrot set is that their forms possess what scientists call "broken symmetry," also described as "similarity with a difference." Here again, we can find a parallel in formal poetry, wherein no two metrical lines are ever identical. Metrical substitutions are one source of asymmetry; but even when no variations occur, subtle differences of stress and duration will make any two lines as different as two snowflakes. There is also a symmetry-breaking tension in the subtle differences between a line's abstract metrical pattern and the speech rhythms playing against it. Rhyme is yet another source of "similarity with a difference," since by definition it's produced by words compounded of both similar and dissimilar sounds.
Finally, we must consider the emergent features which appear in all kinds of dynamic nonlinear systems, from trees to poems. One such emergent feature is beauty. Though Joyce Kilmer claims never to have seen "a poem lovely as a tree," others have found in poems an equal, or surpassing, beauty. Poems not only share the wild, inexplicably beautiful proportions of natural forms, they also engage our minds and perceptions at higher levels, mirroring our moral and intellectual complexities. They have the power to move and transform us.
Consider the following poem by Charles Martin, which illustrates much of the above.
Metaphor of Grass in California
The seeds of certain grasses that once grew
Over the graves of those who fell at Troy
Were brought to California in the hooves
Of Spanish cattle. Trodden into the soil,
They liked it well enough to germinate,
Awakening into another scene
Of conquest: blade fell upon flashing blade
Until the native grasses fled the field,
And the native flowers bowed to their dominion.
Small clumps of them fought on as they retreated
Toward isolated ledges of serpentine,
Repellent to their conquerors. . . .
They were like men who see their city taken,
And think of grass--how soon it will conceal
All of the scattered bodies of the slain;
As such men fall, these fell, but silently.
The poem opens with two lines of carefully interwoven vowels and consonants, alliterating "seeds" and "certain"; "grass," "grew," and "graves." This interlocking mesh of sounds is echoed at higher levels in the metrical rhythms holding individual lines together and binding all of them into a collective unity; it's repeated again in the four strenuously discordant off-rhymes which lock each quatrain into place and make each stanza resemble the others in form and structure.
The first stanza introduces the extended metaphor that will be elaborated throughout the rest of the poem, telling how the seeds of grasses that once grew "Over the graves of those who fell at Troy" were introduced into the New World. By using his metaphor of grass, the poet is able to show not only similarities between the conquest of Troy by Greeks and New World natives by Europeans, but to echo both conquests at a lower level in the war between species of grass. The metaphor enables the poet to set all the historical events against the much vaster scale of evolutionary time, with its vast cyclic movements of Darwinian struggle, adaptation, and mass extinction.
With each stanza, the metaphor complexifies and deepens, so that by the time the poem describes the native flowers bowing to the "dominion" of the new grasses and retreating toward "ledges of serpentine," we can't help but think of the devastation and gradual retreat of Native Americans into those badlands called reservations. The chilling similarities evoked by the metaphor are echoed in the violent chiming of the poem's off-rhymes, as well as in the regularly repeating rhythms of its meter. As the poem expands, its large-scale metaphoric and thematic similarities are mirrored all the way down to the level of individual words as the poet writes punningly of how "blade fell upon flashing blade," where sword and grass, men and nature, are reflected in a single monosyllable.
After the personification of the earlier stanzas, the final stanza comes as a shock. Speaking of the defeated native grasses, the poem declares, "They were like men who see their city taken, / And think of grass--how soon it will conceal / All of the scattered bodies of the slain . . ." thus creating a tangled loop reminiscent of M. C. Escher: The native grasses are like men who, in defeat, think of how grass will cover their dead (human) bodies. A further tangling of levels occurs when we consider the food chain the poet secrets into the poem: the grass seeds arrive in the hooves of cattle; cattle eat grass; men eat cattle; when men die, they are eaten by grass; which is eaten by cattle; and so on to infinity.
Switching from the implicit comparison of metaphor to the explicit comparison of simile ("They were like men who see their city taken" [italics mine]), the poet is able to spring a sudden reversal in the poem's last line: "As such men fall, these fell, but silently." That final "similarity with a difference" delivers the poem's moral shock. Yes, we think, grasses are like men, but they fall silently--without tragedy or conscious suffering, in a world devoid of moral responsibility. In that difference lies a whole universe, filled with all the horror and tragic beauty of our lives. Martin's deceptively simple sixteen-line poem captures an almost infinitely complex set of nested scales--from the lowest, sensual level of patterned sounds up to the highest level of ethical and moral feelings.
Perhaps no poet, mathematician, or scientist of the last two hundred years has understood the laws of creation better--or embodied them more brilliantly--than Gerard Manley Hopkins. From his readings in the Medieval philosopher Duns Scotus, his own first-hand observations of nature, and his reflections on poetic form, Hopkins synthesized a surprisingly accurate theory of organic form. Consider the following extract from his Journal, dated February 24, 1875:
In the snow flat-topped hillocks and shoulders outlined with wavy edges, ridge below ridge, very like the grain of wood in line and in projection like relief maps. These the wind makes I think and of course drifts, which are in fact snow waves. The sharp nape of a drift is sometimes broken by slant flutes or channels. I think this must be when the wind after shaping the drift first has changed and cast waves in the body of the wave itself. All the world is full of inscape and chance left free to act falls into an order as well as purpose: looking out of my window I caught it in the random clods and broken heaps of snow made by the cast of a broom.
Not only does Hopkins see the similarity between snow drifts and waves, he also notes that the pattern of their "wavy edges" is similar to the grain pattern in wood and to another fractal pattern: hills on a relief map. Hopkins sees nature's fractal scaling in the smaller waves etched into the larger drifts. Then he makes the most astounding observation of all, defining the principle underlying all such patterns when he writes, "All the world is full of inscape and chance left free to act falls into an order as well as a purpose . . . [italics mine]. A modern Chaologist couldn't have said it better: Chance, left free to act, will fall into an order. Hopkins notes the same order in "the random clods and broken heaps of snow made by the cast of a broom."
This is not to say that Hopkins believed that the intricate forms of nature are produced by chance alone, or that poetry can be written by giving rein to ungoverned impulses. He believed that natural forms are produced by chance combined with natural law, and spent his whole life working out an elaborate system to explain the patterns he descried in art and nature. The word "inscape," which he used in the passage above, was Hopkins' term for the inner design or pattern that causes an object's distinctive shape; he applied the term to a surprisingly wide range of phenomena, from music, paintings, and poems, to trees, clouds, and waterfalls. He also meant the term to suggest something like what Duns Scotus called haecceitas, the individualizing thisness found in everything in nature. Hopkins coined another word, instress, to define the force that upholds an object's inscape. He believed that even the most seemingly random patterns in nature are produced by instress and can therefore be perceived, understood, and artistically rendered. For instance, in a journal entry dated August 10, 1872, he describes the patterns made by withdrawing waves on sand and notes how the eyes ". . . unpack the huddling and gnarls of the water and law out the shapes and the sequence of the running. . . ."
Hopkins also believed it was possible for a reader to "law out" a poem's inscape. In fact, in some early lecture notes, he described poetry as ". . . speech only employed to carry the inscape of speech for the inscape's sake. . ." In a letter to his friend Robert Bridges, he rebukes poets like Ammons and Olson who ignore a poem's inscape by reading only with their eyes, telling his correspondent not to read his poems "with the eyes but with [his] ears. . . ."
Anticipating the work of Frederick Turner, a synthesizing genius of our own time, Hopkins insisted that all the arts, including poetry, had to be performed to be fully experienced. In a letter to his brother Everard (November, 1885), the poet stated that ". . . every work of art has its own play or performance." And after defining how drama, symphonic music, and painting each achieves its performance, Hopkins added, "A house performs when it is now built and lived in. To come nearer: books play, perform, or are played and performed when they are read."
A final poem by Hopkins will illustrate the depth of his understanding and achievement. Published without a title, it is known by its first line.
As tumbled over rim in roundy wells
Stones ring; like each tucked string tells, each hung bell's
Bow swung finds tongue to fling out broad its name;
Each mortal thing does one thing and the same;
Deals out that being indoors each one dwells;
Selves--goes itself; myself it speaks and spells;
Crying What I do is me: for that I came.
I say more: the just man justices;
Keeps grace: that keeps all his goings graces;
Acts in God's eye what in God's eye he is--
Christ. For Christ plays in ten thousand places,
Lovely in limbs, and lovely in eyes not his
To the Father through the features of men's faces.
By now, we hardly need to point out how the poem performs itself, joining its various elements in one organic wholeness. We might begin by noting that it's a Petrarchan sonnet, then observe how the rhyming of the sestet, cdcdcd, contrasts with the envelope rhyme-scheme of the octave--like the shift from one fractal dimension to another in the branching of human lungs. We might also note how the poem's end-rhymes are echoed in rhymes and thick clusters of assonantally chiming sounds within the lines, and how the lush assonance of the poem finds a parallel in its dense alliteration. Taken together, there's not a more densely interwoven texture of sound in English poetry.
But the rhyming doesn't stop at the level of sound: Hopkins also "rhymes" on the higher level of metaphor, first in the flashing of kingfisher and dragonfly, then in the following lines where he rhymes the sounds of stones falling into wells with plucked strings and rung bells. Rhyming sense with sense, he also makes flashing kingfisher rhyme with vibrating strings and bells as each thing deals out "that being indoors each one dwells."
The entire poem celebrates a self-similarity that extends throughout creation, from the distinctive beauty of pebbles and insects, to the higher forms of ethical and moral beauty in people, up to the very highest informing beauty of the creator himself. According to Hopkins, everything from a dragonfly to God not only possesses a distinctive beauty, a special thisness, it selves--a verb, notice, not a noun. This selving is a living process that happens across all levels of being; it is a lived performance. Hopkins declares that the just man "Acts in God's eye what in God's eye he is"; that "Christ plays in ten thousand places" [italics mine]. Everything in creation--from "selving" pebbles and dragonflies to the man who "justices" and "keeps all his goings graces"--is tangled in one self-interfering loop. In man, the creative principle--what Hopkins calls God--achieves a transformative self-consciousness, able to act "in God's eye what in God's eye [it] is." A man is not only nature observing itself, but embodied personhood (like the incarnate Christ); he is God observing Himself. The final three lines of the poem capture this self-reflexive, paradoxical entangledness with remarkable beauty and economy: " . . . For Christ plays in ten thousand places, / Lovely in limbs, and lovely in eyes not his / To the Father through the features of men's faces." Christ plays to the creator, who is also paradoxically himself, through the "features of men's faces," with their distinctive, character-revealing symmetries and fractal wrinkles.
Hopkins enacts his own selving in this poem in part through his distinctive style, nowhere more clearly than in his first line, where he plays out the implications of his metaphor at every level of sound, syntax, and rhythm:
As kingfishers catch fire, dragonflies draw flame . . .
The pattern of "k" and f sounds in the first half of the line is exactly duplicated in the interlocking pattern of dr and f sounds in the second. Running through this pattern of alliteration and binding it together is the assonantal chiming of short a sounds in "as," "catch," and "dragon." The syntax and rhythm of the two halves of the line are also nearly identical:
As kingfishers catch fire, dragonflies draw flame . . .
A slight asymmetry is caused by the extra unstressed syllable of "As" and the ghostly extra syllable hovering at the end of fire in the first half of the line, as well as by subtle differences in the stress and duration of corresponding syllables in each.
A further asymmetry lies in the way the first line departs from the dominant iambic pattern of the poem. This asymmetry appears again most noticeably in the poem's last line, which hovers suspended between three almost equally plausible scansions of its rhythms. The line can be heard as extremely loose iambic pentameter, with an anapestic substitution in the first foot, an iamb with a courtesy accent on "through" in the second, a normal iamb in the third, and a pyrrhic and spondee (with a feminine ending) in the last two; as trochaic hexameter, with five of its six feet falling; and as an oddly symmetrical three-stress line composed of three anapests, with an extra unstressed syllable after each. The alliteration of the three heavily stressed syllables in "Father," "features," and "faces" lends additional weight to this reading.
Given Hopkins' experiments with accentual verse, one might argue that this poem's metrical irregularities are due to its use of "sprung rhythm," an accentual pattern, usually with five stresses per line, developed by the poet from his readings in Anglo-Saxon verse. But the poem's second line is so perfectly iambic that it establishes a clear metrical paradigm, retroactively influencing the way we see the first, which can be scanned as above, with an iamb, pyrrhic, and spondee before the comma; and "headless" iamb, pyrrhic, and spondee after. Though this particular arrangement bears the inscape of Hopkins' peculiar genius, the use of pyrrhic and spondee is common in iambic verse. Matthew Arnold, for instance, uses it in "Dover Beach" to create a line similar to that of Hopkins: "Come to the window, sweet is the night air."
A similar situation arises in the final line, where, in order to get our five stresses, we have to count either the lightly stressed "To" or "through," as well as "men's"--though none receives anywhere near the weight of the stresses in Fathers, features, faces. Within the larger metrical context, however, we can scan the line as above, as loose iambic. If we regard each line as a gestalt, even some of Hopkins' most difficult lines can be scanned as iambic pentameter--though a pentameter with a higher than usual number of stresses due to his frequent spondaic substitutions.
All of which goes to show how holistic a formal poem is, how every line and every individual foot within it affects the reading of every other line and syllable. Though Hopkins substitutes with unparalleled boldness, especially in the sestet, where the lines buckle and strain against the iambic matrix, we have only to compare this poem to others such as "Spelt from Sibyl's Leaves" or "Carrion Comfort" (both experimental "sonnets" with the same rhyme scheme) to see the difference between it and the poet's more thorough-going experiments in sprung rhythm.
In "As Kingfishers catch fire," it seems that the sonnet form itself (that archetypal thing) reaches down to influence the reading of each constituent part by awakening expectations in us for certain rhymes and rhythms. Every fragment of syntax, every complication of metaphor influences every other part of the poem in ways both large and small. It is difficult to conceive how a poem without the regularly recurring rhythms or self-repeating forms of Hopkins' sonnet could produce such complexity of design or richness of interpretation.
Through parallelisms of syntax and similarities of metaphors and sounds, free verse can sometimes attain isolated expressions of self-similarity in its parts and approximations of order in its overall design; but with fewer rules and less feedback to amplify and vary its constituent elements, it generally fails to achieve the same degree of self-similarity and scaling we find in the best formal verse. Drawn into being not so much by a strange attractor as by a series of provisional judgments and mechanical operations such as hitting return and space keys on a keyboard, free verse can only imitate the most superficial aspects of living forms like trees. Hopkins, even as a young man, suggested as much in an undergraduate essay on "Poetic Diction":
The structure of poetry is that of continuous parallelism, ranging from the technical so-called Parallelisms of Hebrew poetry and the antiphons of Church music up to the intricacy of Greek or Italian or English verse.Suggesting that there's a hierarchy of forms in poetry and music, Hopkins calls the "so-called Parallelisms" of free verse merely technical in nature and denies even the Hebrew poetry of the Bible the organic "intricacy" of classical forms.
Paul Lake* A somewhat shorter version of this essay was published in the October/November, 1996, issue of the AWP Chronicle.