fractal criticism

Brain functioning is known only in broad outline, but its complex nature and multiple feedbacks suggest that its style of information processing is just what poetry exploits. The analysis offered below is very speculative.


Study of brain functioning has yet to properly account for language itself, let alone aesthetic language. But we know that cerebral processes are interlinked, employ multiple feedback, and that consciousness (and so art) could be an emergent property of its complex operations. Not all of our thinking apparatus is located in the brain, and a good deal of human activity is unconscious, not readily imparted by book-learning or encompassed by logic. Synaesthesia has a biological explanation, and rhythm — speech rhythm and metre — is instinctive to us. More than that it is hardly possible to go, except in two areas: metaphor theory and the fractal self-similarity of art.

Fractals are geometric shapes whose parts are themselves reduced copies of the whole. They show self similarity. A fractal may appear as an innocuous zigzag line {1} but zoom in with a magnifying glass and each straight section dissolves into a finer zigzag line. That finer zigzag line may show a yet finer structure, and so on. The process can end at some level of magnification, or the fragmentation go on forever, when the zigzag line would have a dimension neither that of a line nor an area but something perplexingly between the two. Yes, such lines exist, and are indeed very common in nature.

Fractals occur in complex or nonlinear systems. Such systems characteristically have feedback, and each successive state depends on the previous state. We can model a nonlinear system with a simple equation, say Y=AX (X+B), where A and B are constants, which gives us a simple curve with Y steeply increasing as X increases. But nonlinear systems are very different. Because each value of X depends on the previous value of X (and we will represent these two values, present and previous, as X and x, likewise Y and y) we have to write equations linking X and x and Y and y. One example would be: X=x2 + y2 + A. And Y=2xy + B. This is the famous Mandelbrot set, with very remarkable properties. For short distances the plot of X against Y indeed produces a simple curve. But then all changes. For the smallest increase of X, the value of Y is wildly different. Thereafter, for a range of X values, Y plots all over the place — i.e. apparently randomly, chaotically. Then, quite suddenly, when a certain value of X has been reached, the randomness disappears and a simple curve takes over. And in every direction the process repeats in stunning complexity.

And there is a good deal more. Zoom in on the plots (i.e. examine in smaller steps, or increments of X), and each area dissolves into further areas of curves and chaos. Model in even smaller increments, and an even finer structure appears. Or does not appear. With computers we can model the fineness of the increments needed to reach stability and show this fineness by colours on the VDU screen. And the result is a fractal geometry of self- similarity. The blobs and swirls of the whole Mandelbrot structure are repeated at every level of detail. Continue the zooming and the characteristic patterns of linear and nonlinear behaviour constantly appear, shift, dissolve and reappear.

But these patterns are not precisely repeated. There are the subtle variations which are characteristic of complex, nonlinear systems. The X and Y plots proceed smoothly for a while, and then suddenly bifurcate, dissolve into areas of random behaviour, or cycle about points known as "strange attractors". Mechanical bodies (pendulums, weights on springs, etc.) commonly show oscillation or circular motion, but complex motions are not circular repetitions but patterns never exactly repeating themselves. And when these patterns are examined, they very often show a finer structure, indicating that the behaviour of quite simple mathematical expressions can be very complex indeed. {2}

Feedback is endemic to living creatures, and complex systems have been recognized in human beings at every level, from brain functioning to social behaviour. {3} Fractals have also appeared in art theory, not simply as applications in computer art, but in attempts to understand our creative and aesthetic responses. Instinctive human preferences may have an origin in complex mathematical relationships. {4} In poetry we notice that a particular phrase or line appears particularly striking, and it may be that, just as a tuning fork feeds on the vibrations at its characteristic frequency, the particular phrase or line draws its power from its surroundings. Poems often need shaping so that they can resonate all of a piece, and this may explain why a small, finely-crafted piece is often more effective than one longer, richer but more unwieldy.

But is this resonance more than a figure of speech? We don't know. Painters do not paint what they see, but build something from their visual responses using an inherited craft: compositional devices, subtle mixtures of complementary and analogous colours, modification of hue, value and intensity, and so forth. Poets also have their craft, their "rules" of diction, content, imagery, prosody, rhyming, and stanza-shaping being no more than traditional methods of getting a poem to work properly. A poem may expire under too heavy a burden of rules, or individual rules may conflict, requiring some choice, balance and commonsense. But whatever the rules may be, they survive because they are useful, and are useful because they continually relate the part to the whole, accommodate the individual word etc. to the reader's overall expectations. Among these expectations is sense, patterning, a phrasing that seems apt and vivid in its particular context.

How is that patterning achieved but by some features of complex systems? The self-similarity of fractals, the regularity that suddenly appears out of and draws on the chaos that surrounds the poem on every side, and the resonance of individual lines and phrases that echo through the structural matrix of the work can all be applied to the poem under consideration.

There is one further matter. Beginning artists are continually instructed to simplify: to seize on essentials and render those as directly as possible. Sparseness is equally a virtue of writing, not least in poetry. Simplicity also underlies the fractal world. For all the astonishing fecundity of its fractal plot, the equations of the Mandelbrot set are very straightforward, and other fractal patterns may have an even simpler mathematical expression. But each single plot on the VDU screen, each Y intersection with X, is a single representation of the underlying equation, and therefore retains the power to generate the total picture. It resembles the hologram where even a small part of the image has information on the whole, or the virus which takes over the DNA of the cell and starts issuing instructions for its multiple replication. The relevance to poetry? Poems commonly start with an odd phrase or line, which subsequently generates the whole poem — even if sometimes rejected at last, so that the originating words remain unaltered, like a catalyst at the end of a chemical reaction. What gives these words such power that every sensible poet carries a notebook to catch them as they occur?

Consider an analogy. If a tree in a forest is struck by lightning and bursts into flame, what is the probability of the fire spreading? It depends on several things — on the fractal pattern of trees in the forest, on the probability that the tree concerned is a member of the largest cluster, and on the probability that the cluster connects to other clusters. Exact solutions are difficult, but computer simulation indicates that the forest fire burns longest when the tree concerned has a 59.3% probability of belonging to a larger cluster. {5} And this is counter-intuitive, not to be expected. But the suggestion (to the extent that the power of words is anything like comparable) is that key words are most effective when not too thickly surrounded by synonyms — i.e. neither predictable nor obscure. Samuel Johnson's remark "Words too familiar, or too remote, defeat the purpose of a poet. " comes to mind. {6} Analogies prove nothing, but it is nonetheless remarkable that questions of literary practice and sensibility may have a mathematical basis — i.e. be less imposed and arbitrary than Postmodernists sometimes suppose.

Harold McWinnie and others have related fractal dimensions to the aesthetically pleasing golden section, {7} which can of course be derived by other mathematical routes. George Birkoff (1884-1944) suggested an aesthetic measure equal to Complexity / Order, which can indeed be applied to music. The calculations indicate that listeners should enjoy compositions where the succession of notes is neither too predictable nor too surprising, which seems to be the case. {8} But can fractal measurement be extended to poetry? Linguistics is currently a battlefield of contending theories. Brain physiology is known only in broad outline. Experimental aesthetics has not enjoyed the success once expected. And the mathematics of chaos, even with computer iteration, is hard going. But interesting papers are now beginning to appear on the Internet, and prospects may be clearer in a few years' time. {9}

Self Similarity: Themes

The Architects

But, as you'd expect, they are very
Impatient, the buildings, having much in them
Of the heavy surf of the North Sea, flurrying
The grit, lifting the pebbles, flinging them
With a hoarse roar against the aggregate

They are composed of — the cliffs higher of course,
More burdensome, underwritten as
It were with past days overcast
And glinting, obdurate, part of the
Silicate of tough lives, distant and intricate

As the whirring bureaucrats let in
And settled with coffee in the concrete pallets,
Awaiting the post and the department meeting —
Except that these do not know it, at least do not
Seem to, being busy, generally.

So perhaps it is only on those cloudless, almost
Vacuumed afternoons with tier upon tier
Of concrete like rib-bones packed above them,
And they light-headed with the blue airiness
Spinning around, and muzzy, a neuralgia

Calling at random like frail relations, a phone
Ringing in a distant office they cannot get to,
That they become attentive, or we do — these
Divisions persisting, indeed what we talk about,
We, constructing these webs of buildings which,

Caulked like great whales about us, are always
Aware that some trick of the light or weather
Will dress them as friends, pleading and flailing —
And fill with placid but unbearable melodies
Us in deep hinterlands of incurved glass.

C. John Holcombe   |  About the Author    | ©     1997

We start by arranging lines and phrases under themes: —


past days
Sea the heavy surf of the North Sea
great whales

Land and Sky

hoarse roar
the aggregate / They are composed of
the silicate / Of tough lines
the cloudless almost / Vacuumed afternoons
lightheaded with a blue airiness
the light or weather


whirring bureaucrats
settled with coffee in the concrete pallets
Of concrete like rib-bones packed above them
These / Divisions persisting
Caulked like great whales
Us in the deep hinterlands of incurved glass


constructing these webs of buildings Illness lightheaded
a neuralgia
like frail relations
pleading and flailing

Whereas mythic criticism examines the imagery and its power, fractal criticism is simply concerned with self similarity of themes. We want to know what themes arise, and how each is developed. In each appearance we would hope to see a variation in miniature of the poem as a whole. Let's start with the most obvious: control.

Though heavy, oppressive and consistently invoking the natural world, in what way does control repeat the overall theme of the poem? The title is "The Architects" and, although these use concrete and other obdurate building materials, and may even"caulk" damp-courses etc., control is not the usual image of the profession. Architecture is one of the creative arts, and foundation courses start in art colleges. Is the title a misnomer? Possibly. It may have been chosen to emphasize the "we" that appears so abruptly in line 23, or in default of something more abstruse. A more exact but materialistic title could have been "The Uses of Silica".

But perhaps the theme really is architects, the way their imaginations are curbed, not only by the practicalities of building but the constraints of cost and client preferences? The imagery of the third stanza would support that interpretation — a busy architects' office closely resembling the usual bureaucrats' — and the final stanza indeed suggests that their original conceptions have been imprisoned in some friendly but uncomfortable appearance, or lost in the "hinterlands of incurved glass." But if this is so, then the first two stanzas lead awkwardly into the theme, and need attention.

Also preponderant are the themes of "land" and "sky". Weather, the tough silicates of rocks and the vertigo of high buildings on clear afternoons make their appearance. These clearly are not the theme of the poem announced by the title, and the observation that we are all imprisoned /controlled by the constant features of the landscape is too obvious to be worth elaborating. But suppose the poem was indeed about control, only the control was self-exerted? Bureaucrats follow the time-wasting and sterile procedures of office-dwellers: architects yearn for more freedom but know the constraints of their profession; the elements of the natural world follow their ineluctable cycles of geological creation, weathering and erosion.

Suppose we leave that as a possibility, and look at the other themes. That of "creation" belongs to the architects, and appears only once. That of "time" appears in the aberrant opening stanzas. And that of "illness" applies to the architects only: the bureaucrats are self-absorbed and content with their routines. Is that the essence? The poem is called "The Architects" because they alone are aware of the domination by the inherent nature of things? Again, if decided upon, the poem needs tidying up, with the themes more clearly established.

Self Similarity: Imagery

Matters are more straightforward with imagery. All is drawn from the natural world, and appears hard, inescapable and indifferent to human interests. The sea is invoked with the action of surf on grit and pebbles. The sea cliffs are burdensome, and the silicate nature of the aggregates — hard and polished — has entered the lives of bureaucrats. The afternoons are empty (vacuum-cleaned) and seem oppressed by the weight of floors packed like rib-bones above them. The submarine haunts of the whale are continued in the glass hinterlands, and neither whales nor architects are happy with their surroundings. The only matters that need attention — since the imagery of the poem is consistent, both as a whole and in the individual phrases — is the "impatient" of line 1, the "whirring" of line 11 and the "webs" of line 25.

Risk Taking: Closeness to Chaos

Complex systems develop areas of stability, or quasi-stability, on the very edge of chaos. {10} That is one of their characteristics, what makes them interesting. The Modernists championed the new, and the Postmodernists focus on the socially antagonistic and iconoclastic. But the suggestion from complex systems is that art is important for the patterning it creates from chaos. It is not the order nor the chaos per se that are themselves important, but their arrangement in a particular artwork. That order or regularity grows out of the chaos. It feeds on it, and cannot be meaningful otherwise. For this reason blueprints do not exist for artworks: there are no moulds into which content or technique may be simply poured. The greatest art is that which takes the largest risks, and is the more encompassing, but the need is still for patterning, balance, variety in order. Anyone can petulantly tear up the beliefs of the previous generation, but only a great artist can embody them in new ways and make that conception more detailed and far-reaching. The west's passion for novelty, and the east's deep respect for tradition may not be antagonistic, though the balance and expression differ with the cultures concerned.

Poems therefore have to be fought for, and are continually asserting themselves against the obscure, the incoherent, the dark forces of our instinctive natures. The greatest poems are not necessarily made from the most obviously felt emotions — often the reverse since a poem can be overwhelmed by the originating experience, and so stay incoherent and uncommunicated — but they are made from deep strands of intellectual and emotional instability. Their appearance is often a wonder, and their completion a revelation. "I had no idea I really felt that," is a common response.

If that is so, we should be able to feel the poem organizing itself, pulling phrase after phrase into order and coherence. And the craft and devices used should be appropriate, mediating with the surrounding incoherence, but not smothering it. The trouble with "greeting card verse" is not that the themes are trite, or the style hackneyed, but that the material is predictable, predigested, in no way vital or threatening. What is said may be "true", and charmingly expressed, but the piece remains verse: innocuous, without intellectual or emotional charge.

What incoherence is being struggled with in this poem? Quite a lot, though not always successfully. Something of the elements which created the concrete aggregate apparently lingers on in the buildings, but "impatient" is not the word to describe their continuing influence. Indeed the energy of the North Sea is too strong for the buildings, which are not sufficiently stressed. The first stanza should be rewritten, or entirely rethought. The energy of the natural world contrasts with the mechanical actions of the bureaucrats and the constrained lives of architects and whales, but why is that contrast important? Why is nature antagonistic to man? What is missing between the savagery of the North Sea with which the poem opens and the gentle melancholy with which it ends? Something is started with the word "underwritten" of line 7, but thought remains incomplete. Similarly the jump in line 23 from "they" to "we" — presumably from bureaucrats to architects — is arbitrary as it stands, and weakens whatever it is that "attentive" refers to.

Is the poem organizing itself? Sometimes. But look again at lines 7 to 10. Are the words drawing order out of chaos or simply following a train of association? Certainly they don't sink intellectual claws into the scene. And in lines 14 to 15, if the bureaucrats are turning away from outside experience, what is the troubling "generally" attempting to achieve? The thought here becomes very thin, no doubt introducing the light-headedness that comes with high-rise offices, but weakening the overall tenor of the poem. The bureaucrats are muzzy-headed, the architects are talking about divisions, and the whales are pleading and flailing: in what ways can any of these be said to have created insight and order from the confused indifference of the natural world? Some clarification is needed.

Resonance: Drawing on the Whole

Engineers are familiar with resonance. All systems have a frequency at which they naturally vibrate (or build up rapidly in amplitude if electrical). At some distance from the natural frequency the effect of inducing a vibration is very small, but this builds dramatically as the natural frequency is approached. For that reason, troops break step when marching across bridges, in case their rhythm should coincide with that the bridge, and so bring the whole structure down. And for that reason as well, to extend the analogy, the exact word is unusually effective in literature, hanging in the mind long after we have finished reading.

But perhaps literary resonance is not mere analogy. By their intellectual cohesion, their emotive buildups, their use of metre and a host of other devices, poems may in fact operate like complex systems, with various nodes and points of maximum activity. Complex systems are indeed complex, with a behaviour not easily explained by their constituent parts. Often the mathematics is so difficult that solutions are more readily obtained by experimentation on models rather than by solving the equations concerned, even supposing that the equations can be adequately derived. Exactly the same applies to poetry. Much of it is built by experiment, by trial and error. It is well known — indeed is a feature of a well-constructed poem — that a word must fit neatly as though predestined into a particular line, and that this one word will have repercussions throughout the whole poem. The words all then appear inevitable, drawing on each other for their intellectual content and emotional support. And those which fail, do so spectacularly. What do we think of whirring, webs of buildings, and deep hinterlands of incurved glass?

Whirring suggests something hidden, mechanical, perhaps ineffectual, and so not inappropriate. Yet the word seems wrong, perhaps because it is unsupported by the associations of the words around. All are tough, obdurate, heavy, inert, whereas whirring seems lightweight and fretful. There is no alliteration or onomatopoeia to settle the word, or indeed other devices to fasten it into the taxis (structure) of the poem.

Similar objections can be levelled against these webs of buildings. The intent is obvious enough — to link buildings with architect and inhabitants — but the link simply states what should emerge from the context: it is applied, imposed, manhandles the meaning too brutally.

And finally there is deep hinterlands of incurved glass. Of course deep is acting as a bridging mechanism, linking the submarine world of whales and the supercooled liquid that is glass with psychological (incurved) constraints on our lives, but should so far-reaching or contentious a suggestion be introduced in the final line without prior development or warning?

Conclusions: Suggested Improvements

The following seem necessary:

  • Rewrite the first stanza.

  • Reshape the second stanza to lead into the third better.

  • Clarify who is who in fourth and fifth stanzas.

  • Replace whirring, these webs of buildings and deep hinterlands of incurved glass.

Some of the shortcomings have been corrected in a new version, now entitled Office Workers.


1. Benoit Baald at the Birdhouse. Jan. 2004.
2. John L. Casti's Complexification: Explaining a Paradoxical World Through the Science of Surprise (1994), Roger Lewin's Complexity: Life at the Edge of Chaos (1992), and Edward Lorenz's The Essence of Chaos (1994). Mathematicians should read Denny Gulik's Encounters with Chaos (1992) or K. Falconer's Fractal Geometry: Mathematical Foundations and Applications (1990). Fractals for the Classroom (1992) by H.O. Peitgen et al is a simpler treatment.
3. Jack Cohen and Ian Stewart's The Collapse of Chaos: Discovering Simplicity in a Complex World (1994).
4. Harold McWhinnie's Chaos, the Brain and the Arts. Artery International. 1999-2000. NNA. Aug. 2002.
5. Greatly simplified. The fractal pattern is that of a Sierpinski gasket, with a self-similarity (fractal dimension) of 1.585. See Chapter 7 of Peitgen et al 1992 for a fuller treatment.
6. Samuel Johnson's Life of Dryden in Lives of the Poets (1781).
7. Summarized in McWinnie 1997.
8. pp. 10-112 in Manfred Schroeder's Fractals, Chaos, Power Laws (1992).
9. For tentative beginnings see: N. Katherine Hayles' (Ed.) Chaos and Order: Complex Dynamics in Literature and Science (1991) and Karl Frank's Notes Toward a Strange Attractor: Wallace Stevens' Anticipation of Scientific Chaos Theory (1995).
10. I. Prigogine and I. Stengers's Order Out of Chaos (1984).

Internet Resources

Few of these sites relate to literary criticism, but they do give some idea of the importance and potential applications of complex systems.
1. Complexity and Chaos Theory.
. Listings on the System Thinking Press (commercial) site: a wide range of applications.
2. Complexity, Complex Systems & Chaos Theory. Commercial site with extensive listings.
3. New England Complex Systems Institute. Applications in research and education.
4. Complexity Society. UK society with journal and online digests.
5. Fractal Visual Poetry. 3 Apr. 2006. Kaz Maslanka's blog on mathematical poetry, here analyzing a fractal poem by Marko Niemi (closer to concrete poetry, but fascinating).
6. WWW Resources on Nonlinear Dynamics and Complex Systems.
. Listings at NCSL, Pohang, Korea.
7. Some Literary Criticism quotes. Tim Love. Dec. 2003. Extensive listing: includes note on Birkoff's aesthetic formula.
8. An Aesthetic Consideration of the Fractal Dimension: Implications of chaos theory for aesthetic measure studies. Hal McWhinnie. Note on the possible connection between the golden section and the Hausdorff dimension.
9. Beauty and the Beast: The Role of Aesthetics in Economic Theory. Cassey Lee and P.J. Lloyd. 2002. Notions of beauty in mathematics and economics, quoting Birkoff's paper.