The discussion here centers on how self-similarity, or parts resembling a whole, has been a recurring aspect in the most diverse fields of human thinking from antiquity to the present. Primary examples are drawn from physics, biology, cybernetics, alchemy, philosophy, myth and, of course, language and literature. The author argues that self-similar patterns are one of the persistent ways in which the human mind structures its experience and knowledge of the world, and then examines some of the questions that arise when an almost ubiquitous concept, structure or linguistic and literary phenomenon resurfaces in a new guise in hard science.

Of Parts and Wholes: Self-similarity and Synecdoche in Science, Culture and Literature

Dirk Vanderbeke

<1> Over the last decades the concept of self-similarity has gained some prominence in fractal geometry and in the field of nonlinear dynamics or deterministic chaos, a.k.a. chaos theory. It indicates that there is a recursion of pattern within pattern, a symmetry across scale. In his seminal book The Fractal Geometry of Nature Benoit Mandelbrot offers the following definition: "When each piece of a shape is geometrically similar to the whole, both the shape and the cascade that generate it are called self-similar" (Mandelbrot, 34 a-b, italics in the original; for the mathematical definition cf. Mandelbrot, 349 b-350 a). Fractals are complex regular or irregular geometrical shapes that are characterized by a 'broken' dimension, i.e. they are not one, two or three dimensional but something in between [1]. One of the most famous examples is the Koch Snowflake -- with a dimension of 1,2618 -- in which ultimately very part resembles the whole.


Figure 1: Koch Snowflake (from Mandelbrot 43).

<2> The Koch Snowflake is an abstract construction of a strict form of self-similarity. In other cases the similarity appears only at certain levels, as for example in some of the equally famed Mandelbrot sets (cf. Mandelbrot, 188-9; for some of the colorful images cf. David E. Joyce). The important point in self-similarity is that in a geometrical structure, in a mathematical set or in other phenomena every part or at least one part resembles the whole. This principle has a distant analogy in language. In the discussion of 'literature and science,' the metaphor in scientific language has for a long time been a special point of interest. I want to introduce another trope that I consider to be almost as important, but for very different reasons. It is synecdoche, which according to George Lakoff's and Mark Johnson's Metaphors We Live By is "part of the ordinary, everyday way we think and act as well as talk" (Lakoff and Johnson: 37). Synecdoche is also called pars pro toto, i.e. in speaking about a part of something we actually mean the whole. Examples are the head for a person when we count an assembly of 200 heads, or the sail for the ship when we talk about a fleet of 50 sails. This is the most common case of synecdoche; it is merely a way of speaking and of little relevance for the following pages. But synecdoche also frequently tends to intrude on our perceptions or concepts of the world, i.e. various phenomena are perceived in a way so that the part is really similar to the whole. I want to call those cases 'true synecdoches' [2]. The first suggestion that the synecdoche might be more than a way of speaking is almost 300 years old; it can be found in Giambattista Vico's New Science where he writes that synecdoche only developed into metaphor when "particulars were elevated into universals or parts united with other parts together with which they make up wholes" (Vico, 130, § 405).

<3> In this paper I want to show how the idea of self-similarity or parts resembling a whole has been a recurring aspect in the most diverse fields of human thinking from antiquity to the present -- which means that I will offer a lot of examples in order to prove my point. The examples will be from physics, biology, cybernetics, alchemy, philosophy, myth and, of course, language and literature -- and it will be necessary to jump from the present to the past and back in order to follow up on specific motifs. I want to argue that self-similar patterns are one of the persistent ways in which the human mind structures its experience and knowledge of the world. And then I want to touch upon some of the questions that arise when an almost ubiquitous concept, structure or linguistic and literary phenomenon resurfaces in a new guise in hard science.

The physical world

<4> I have already mentioned self-similarity in chaos theory. But it is not the only case where we find it in modern physics. In his book Wholeness and Implicate Order the physicist David Bohm presents his theory of an undivided wholeness of being in the universe. To illustrate this idea he uses the image of a hologram record, writing that "the key feature of that record is that each part contains information about the whole object" (Bohm, 177, italics in the original) and that "the form and structure of the entire object may be said to be enfolded within each region of the photographic record" (ibid., italics in the original). The metaphor of a hologram thus offers the concept of a universe in which every part contains information about the whole. In addition to the hologram, Bohm also uses the image of a seed and a tree -- and he adds: "In terms of the implicate order one may say that everything is enfolded into everything" (ibid.). This image of a universe in which each part potentially enfolds the whole has a long history, and Bohm actually points out that the idea and its terminology were already to some extent present in the medieval philosophy of Nikolaus Cusanus (cf. the interview in Davies and Brown: 122). Nikolaus had already used the terms of explicatio and implicatio, i.e. an unfolding and enfolding of events in time, even if he saw God as its origin, and not the physical world. And indeed aspects of self-similarity are present in Nikolaus's philosophy, for he already claimed that everything partakes of the whole [3]. Moreover, Bohm's phrase of everything being enfolded in everything else recalls the pre-socratic philosopher Anaxagoras and his far older dictum that "any part is a mixture in the same way as the All, on the ground of the observed fact that anything comes out of anything" (quoted from Aristotle, Physics, Bk. III, 4). And it may not come as a complete surprise here that Nikolaus Cusanus also affirmed Anaxagoras' statement [4]. David Bohm's theory of the basic principles underlying the physical universe thus not only employs aspects of self-similarity but also resurrects philosophical concepts with a long pedigree.

<5> On a less abstract level self-similarity had already made its appearance in physics at the beginning of the 20th century. Nils Bohr, in his Nobel Prize in Physics Award Address 1922 said about the structure of the atom: "In this picture we at once see a striking resemblance to a planetary system, such as we have in our own solar system" (Bohr, 316). Bohr also points out the differences (cf. 318-9), but quite obviously the analogy presented itself, and Max Born said of Bohr's theory that it performed "a great magic; indeed its form is rooted in the superstition (which is as old as history of thought) that the destiny of men could be read from the stars" (quoted from Miller, 114-5). Actually, the problem of the atom and its stability very much resembled those that early modern astronomers had to face when they discovered that the planets moved around the sun without eventually spiralling in even though no recognizable force was working on them. Again we find a model that is built on an analogy between phenomena of vastly different scales.

<6> But then the idea of a minuscule solar system or galaxy is probably familiar to everyone who has ever discovered a universe in the nail of the left little finger after the injudicious use of restricted substances. But here I leave physics and move on to some less scientific examples of a self-similar universe.

<7> The motif has recently been taken up in the film Men in Black, which ends with a special effect sequence in which the camera seems to rush away from earth until it shows first the solar system, then the galaxy and finally the universe -- which turns out to be merely a marble in the game of some super-universal alien. In addition, the film features a tiny galaxy set in a jewel in the collar of a cat, and thus self-similarity works in all directions. But then the idea of a world contained in a jewel may actually go back to one of Margaret Cavendish's poems, "A World in an Eare-Ring" [1653]:

An Eare-ring round may well a Zodiacke bee,
Wherein a Sun goeth round, and we not see.
And Planets seven about that Sun may move,
And Hee stand still, as some wise men would prove.
And fixed Stars, like twinkling Diamonds, plac'd
About this Eare-ring, which a World is vast. (Cavendish, 253)
In another poem she writes "Of Many Worlds In This World" [1653]:
Just like unto a Nest of Boxes round,
Degrees of sizes within each Boxe are found.
So in this World may many Worlds more be,
Thinner, and lesse, and lesse still by degree;
Although they are not subject to our Sense,
A World may not be bigger than two-pence. (Cavendish, 252)
At the bottom of this form of self-similarity we encounter the old concept of a resemblance between microcosm and macrocosm, a concept most famously employed by medieval and renaissance philosophy and especially alchemy. One of the most basic rules of alchemy was 'as above so below,' and the English physician Nicholas Culpepper wrote in 1654:

If you do but consider the whole universe as one united body, and man an epitome of this body, it will seem strange to none but madmen and fools that the stars have influence upon the body of man, considering he, be[ing] an epitome of Creation must needs have a celestial world within himself. (Nicholas Culpepper, Pharmacopoeia Londoniensis: or the London Dispensatory [1654]; quoted from Thomas, 398)
Moreover, not only does the human body resemble the cosmos, it also finds its analogy in the natural phenomena of our planet. In consequence, the winds are regarded as the planetary breath and earthquakes are equal to digestive problems or fevers (cf. Gebelein, 201). The similarities exist on more than two levels, from the cosmos to the planet, the individual and further downward as the poems by Margaret Cavendish show, until, ultimately, the human body is seen as inhabited by a multitude of life forms. This idea, of course, invites parody. In Rabelais' Gargantua and Pantagruel (bk. II, ch. 32) the narrator explores a country with large cities inside the mouth of Pantagruel, and Mandelbrot, in his discussion of self-similarity, mentions a few lines of Swift's "On Poetry: A Rhapsody" which claim:
So, Nat'ralists observe, a Flea
Hath smaller fleas that on him prey,
And these have smaller fleas to bite 'em,
And so proceed ad infinitum. (Swift, ll.337-40, cf. Mandelbrot, 402 a)
But then the concept of microcosm and macrocosm occasionally also served scientific research, as in the case of William Harvey, whose anatomical work was based on the attempt to affirm the analogy between the upper and the lower regions. After he had discovered the circulation of blood in the body, he wrote: "The blood motion we may be allowed to call circular, in the same way as Aristotle says that the air and the rain emulate the circular motion of the superior bodies" (quoted from Mandelbrot, 149 a). Blood circulation is thus analogous to the meteorological phenomena of our planet, which again resemble the movement of the heavenly bodies.

<8> This similarity between the part and the whole, the microcosm and the macrocosm, the human body and the world, has a long history and goes back to the earliest creation myths, in which the heavens and the earth are created out of the body of a giant being. In the Poetic Edda, the giant Vafthruthnir tells Odin about the creation of the Earth:

Out of Ymir's flesh | was fashioned the earth,
And the mountains were made of his bones;
The sky from the frost-cold | giant's skull,
And the ocean out of his blood. (Poetic Edda, "Vafthruthnismol," stanza 21)
A similar account is offered in the Babylonian Enuma Elish after Marduk has slain the monster Tiamat:

Then the lord rested, gazing upon her dead body,
While he divided the flesh [of the ... ,] and devised a cunning plan.
He split her up like a flat fish into two halves;
One half of her he established as a covering for heaven. (Enuma Elish, "Fourth Tablet," ll. 135-8)
Further examples can be found in Joseph Campbell's psychoanalytically informed interpretation of world mythologies The Hero With a Thousand Faces (part II, ch. 1.5) -- it seems as if the world-as-body image is a stock item of creation myths.

<9> From this point it may be possible to set out on a first attempt at explanation. It has been suggested that the narratives of myth offer themselves to the exploration of ancient knowledge and "the point where myth and science join" (de Santillana and von Dechend, vii). And in the discussion of this aspect of myth we come across a point we have met before. In their book Hamlet's Mill. On the Origin Of Human Knowledge and its Transmission Through Myth, Giorgio de Santillana and Hertha von Dechend write: "In the archaic universe all things were signs and signatures of each other, inscribed in the hologram ..." (ibid, 343). On the other hand, Campbell points out that the mythical cosmogonies are analogous to human procreation and birth (cf. Campbell, part II, ch. 1.4); moreover, the imagery of creation myths has frequently been related to the development of infants [5], and the perspective of children as akin to the world view of the so-called primitive people (cf. e.g. Freud, 8, 90 and passim). It might be possible, then, to link the concept of a universe, in which everything resembles everything else and ultimately the human body, to the early undifferentiated phase, in which the world appears to the infant as the extension of its body. The creation of the world from the dismembered body of the primordial being would then retell the experience of disintegration at the end of the undifferentiated phase -- an experience which, as Lacan has pointed out, returns in dreams of disjointed limbs and organs (cf. Lacan, 97). But I want to stress two caveats at this point:

a) While it may just barely be possible to find analogies between myth and the experience and world view of children, it is not quite as easy to see the later examples in the same light. And even though it makes some sense to locate the origin of the concept of microcosm and macrocosm in the myth of the world-as-body, it would seem more than a little strange to read Nils Bohr's and David Bohm's ideas as a return of the undifferentiated phase.

b) The argument of myth as akin to infant experience itself employs patterns of self-similarity, for it rests on the assumption that individual birth repeats the creation of the mythical, i.e. anthropomorphized, world and that the mental development of each child resembles the development of mankind -- again the part resembles the whole. But here we move from concepts about the physical world to a similarity between the individual and the community, i.e. the social world.

The social world

<10> In his book Cybernetics, Norbert Wiener writes: "[I]t is certainly true that the social system is an organization like the individual that is bound together by a system of communication" (Wiener, 24, quoted from Otis, 182 [6]). In this passage Wiener takes up an image that had been around for a while in biology. Here we find the suggestion of a similarity between the small and the large in the concept of a superorganism, i.e. an organism which is made up of smaller organisms. While this actually exists in the case of colonial hydrozoans or jellyfish like the Portuguese man-of-war (cf. Wilson, 383-6), it has also been claimed that social insects, in particular ants, form superorganisms. William Morton Wheeler first suggested this idea in 1911, and in their seminal book on Ants Bert Hölldobler and Edward O. Wilson write: "[Wheeler] stated that the animal colony is really an organism and not merely an analogue of one" (Hölldobler and Wilson, 358). In the meantime this idea has spread and now can also be found in various other places, e.g. in Douglas Hofstadter's Gödel, Escher, Bach or in A.S. Byatt's novella Morpho Eugenia. In Morpho Eugenia this concept is then transferred from social insects to life in a Victorian mansion, and thus a human community appears as a superorganism that is made up of smaller organisms. In recent times the city has been depicted as a superorganism (cf. Johnson).

<11> But then the human community has been described as a kind of organism for a long time. Thomas Hobbes Leviathan is only one of the better-known texts that use this image. He describes the state as an artificial man with the sovereignty as its soul and each institution in analogy to some organ [7]. Then again we can find the individual and its development from infancy to maturity and ultimately death as a model for the history of nations. Vico's New Science is to some extent based on this analogy (cf. for example Vico 75, § 211-219, in which Vico links the linguistic development of children to the earliest history); it is a special case of the idea mentioned above that the individual psychology is linked to the psychology of mankind and in particular the infant world view to the concepts underlying myth and the religious -- and in particular animistic -- practices of the so-called primitive peoples. Then again the hypothesis that ontogeny recapitulates phylogeny, that individuals, in their embryology and growth, repeat the adult forms of their ancestry, once more related the individual to historical developments on a grand scale, and embryonic growth was seen as similar to the evolutionary history of the species [8]. In all these examples we find a similarity between the single unit of a collective and the whole, either in the performance of the 'social body' or in development over various time spans.

<12> Again, this principle can be extended in both directions and the individual body has occasionally been depicted as a community. In his book The Selfish Gene, Richard Dawkins explores the idea that our genes are the agents of survival and each organism is just the tool to create new genes (cf. Dawkins, 19-20 and passim). One of the metaphors he presents is a boat race:

The oarsmen are the genes. The rivals for each seat in the boat are alleles potentially capable of occupying the same slot along the length of the chromosome. Rowing fast corresponds to building a body which is successful at surviving. The wind is the external environment. The pool of alternative candidates is the gene pool. (Dawkins, 38)
But the image of selfishly competing replicators was not really new when Dawkins used it in his book. John Barth in his short story "Night-Sea Journey" introduced a sperm cell as the narrator, who finds "abhorrent and tautological the doctrine of survival of the fittest" (Barth 1978, 6), but actually proves to be "the sole remaining swimmer" (ibid.) in the race towards biological destiny and conception. In his novel Tidewater Tales Barth then adds the complementary image of egg cells, appropriately named April, May and June, on their way through the ovaries to meet the swimmers (cf. Barth 1988, 145-161, 371-93 and 619-632). The latest evocation of the body as a community I have found is from Jonathan Franzen's novel The Corrections:

In the morning the blood was crowded with commuters, the glucose peons, lactic and ureic sanitation workers, hemoglobinous deliverymen carrying loads of freshly brewed oxygen in their dented vans, the stern foremen like insulin, the enzymic middle managers and executive epinephrine, leukocyte cops and EMS workers, expensive consultants arriving in their pink and white and canary-yellow limos, everyone riding the aortal elevator and dispensing though the arteries. (Franzen, 332)
And this seems to be the appropriate place to really move over to the realm of literature.

Literature

<13> The literary examples I have offered up to now were all concerned with an explication of self-similar concepts or metaphors. In this they were not inherently different from the scientific or philosophical texts that offered analogous concepts. But there are also those texts that are actually self-similar, i.e. they contain passages -- occasionally on various levels -- that appear as smaller versions of the text which contains them. These, of course, would be subsumed under the denomination 'synecdoche' or pars pro toto in literary criticism, and thus the analogy with self-similarity seems to have escaped notion. Recent examples are Stanislaw Lem's Fiasco, Thomas Pynchon's novels, and, once more, A.S. Byatt's Morpho Eugenia. Byatt's novella contains a fable that retells the whole text. Similarly Lem's book about a disastrous expedition to a foreign inhabited planet in one of its chapters offers a passage from a novel about an expedition into Africa that ends in the same destructive manner as the future first contact story (cf. Lem, 94-102). Moreover, a multitude of allusions to the myth of Orpheus and the motif that the desire to see brings death adds another level of self-similarity to the text. And, finally, Pynchon's books are riddled with the letter V on so many levels that any further elaboration seems superfluous (cf. Vanderbeke, passim).

<14> These examples are from texts published after the advent of chaos theory; moreover, they all demonstrate their authors' interest in science and thus a direct influence cannot be ruled out. But the pattern is certainly older than just a few decades. My favourite example is from a book that can hardly be located in the context of 'science and literature,' i.e. Raymond Chandler's The Big Sleep. On the first page of the novel, Philip Marlowe enters the mansion of his client.

Over the entrance doors, which would have let in a troop of Indian elephants, there was a broad stained-glass panel showing a knight in dark armor rescuing a lady who was tied to a tree and didn't have any clothes on but some very long and convenient hair. The knight had pushed the vizor of his helmet back to be sociable, and he was fiddling with the knots on the ropes that tied the lady to the tree and not getting anywhere. I stood there and thought that if I lived in the house, I would sooner or later have to climb up there and help him. He didn't seem to be really trying. (Chandler, 589)
Anyone who knows the novel will realize that this is the story in a nutshell: a lady with a problem that paralyzes her applies for the help of a shady friend who does not really try to release her from her paralyzed state until finally Marlowe steps in and frees her from the dilemma.

<15> The most accomplished example of a self-similar literary text is probably James Joyce's Finnegans Wake. It is well known that the Wake has a circular structure and that this structure is repeated in its four books, in the chapters, in a multitude of paragraphs and ultimately down to single sentences. The structure of the text appears as a wheel made up of smaller and smaller wheels. This multi-levelled circular movement is at least to some extent the result of Vico's influence on Joyce, and a closer look at The New Science will show that this text also repeats in its structure the pattern it describes. Again we find wheels within wheels on all possible levels. And then there is another book that strongly influenced Joyce, the Book of Kells from the early Middle Ages, in which we find a similar pattern. In some of the elaborate illuminations there are once more circles within circles, and they might well have been used by Mandelbrot as artistic expressions of fractals, as, for example, the so called Chi Rho page [9].


Figure 2: Chi Rho page
<www.faksimile.ch>

More examples could easily be found; quite obviously self-similarity has been around for a long time in literature and the arts, in science and philosophy. The equally obvious conclusion is that this recursive pattern must have a particular significance.

<16> But looking at the examples from science and philosophy it cannot escape notice that in almost all cases the idea was discarded sooner or later. The microcosm/macrocosm analogy has been relegated to esoteric obscurantism, Nils Bohr's first model of the atom was shown to be far too simplistic, the analogy between ontogeny and phylogeny has been abandoned and so has the link between infant experience and the so-called primitive world view, and David Bohm's theory of Wholeness and Implicate Order in the universe is at best an 'also ran' among competing models in quantum theory. In other cases, as for example in the social sciences, in history and in the concept of the superorganism, the idea had mainly a metaphorical function, and more often than not the metaphor lost its usefulness eventually.

<17> A closer look reveals that these concepts frequently make their appearance in speculative attempts to impose some order on the perceived world, as in creation myth, the idea of microcosm and macrocosm or in Bohm's theory of an implicate order in the universe. But they also pop up with some regularity at the initial moment, when an old paradigm needs to be replaced and a new scientific concept is just around the corner. About this time Thomas Kuhn states in his book on The Structure of Scientific Revolutions that: "the scientist in crisis will constantly try to generate speculative theories that, if successful, may disclose the road to a new paradigm and, if unsuccessful, can be surrendered with relative ease" (Kuhn, 86). This is the time when potential theory-constitutive metaphors are offered and explored. And among these speculations recourse is almost invariably taken to self-similar structures akin to the synecdoche, which, as quoted above, is "part of the ordinary, everyday way we think and act as well as talk" (Lakoff and Johnson, 37). Aesthetics play an important part in the course of theory formation, as almost every scientist will admit, and self-similarity is quite obviously a concept that appeals to the scientific and non-scientific beholder. Equally invariably, though, this early speculation is sooner or later abandoned or at least qualified. But it remains present far longer in the public understanding of the sciences and in particular in texts of popular science, as it offers a comparatively easy access via a catchy image and structure that is not only part of our everyday thinking but also a recurring aesthetic principle in the arts.

<18> This leads to a serious problem in the vast field of 'literature, rhetoric and science.' In our approach to the sciences and scientific writing it is necessary to take into account at what moment in the development of the respective science a text was written. In the discussion of 'rhetoric and metaphor in science' this aspect is hardly ever acknowledged. The texts on this topic are usually happy enough to prove that all language, including the language of science, is metaphorical (cf. for example Hayles, 212) and rest their case [10]. But far more often than not, the examples are taken from scientific texts that belong to the early approaches to new concepts, i.e. a time when a multitude of ideas and metaphors spring up, only to be rejected or qualified after some time. This does not challenge the notion that metaphors are an important aspect of science. Theory-constitutive metaphors occasionally stay around for a long time and their exploration can play an important part in the development of scientific thinking. But in the evaluation of metaphors and imagery in science it is of crucial importance to take the internal logic of scientific development into account. The recurrence of specific metaphors -- and self-similarity is only one case among others -- could then provide analysis with a useful tool for the evaluation of scientific texts and their respective place in the history of a science. Occasionally they could even serve as an indication that the speculative stage might not yet be over.

<19> But now self-similarity has reappeared in mathematics and also in chaos theory with its impact on physics, chemistry and biology, i.e. the hard sciences. Literary critics have not been slow in appropriating the resurrection of an old aesthetic image, and this repeats the question of its previous and present significance with a vengeance. It would certainly be rash to simply assume that once more self-similarity is just a phase in the infancy of a science, for now we do have a rigid formalism and thus definitely more than just a metaphor. So, how does it relate to its precursors?

<20> One solution could be the well known 'new age' assumption that there is something like a intuitive grasp of the underlying principles of the physical world in earlier philosophical or scientific concepts as well as in other expressions of human creativity. Thus one might try to score a point in the perennial battle of the two cultures and claim that philosophy and literature anticipated scientific development. But this approach faces a serious internal contradiction. While self-similarity can undoubtedly be seen as an important aspect of the respective concepts and texts, it is quite obvious that in their attempts to present an orderly and hierarchical universe they are far from being anticipations of a fractal or chaotic nature of the world.

<22> A different and more promising approach to the problem alters the question and now asks, what exactly is the relevance of self-similarity in chaos theory and in fractals? The introduction to the topic quoted almost invariably by all literary critics who touch upon chaos theory is James Gleick's popular science book Chaos: The Making of a new Science, and here we find the statement about Mandelbrot's research: "Above all, fractal meant self-similar" (Gleick, 103). This statement is usually taken at face value and frequently quoted or paraphrased without further qualifications. I want to offer only two examples, assuming that anyone who has ever done research in this field will be able to come up with a few others. Helmut Bonheim writes that "fractal essentially means 'self-similar' -- it implies recursion, pattern inside of pattern, 'symmetry across scale'" (Bonheim, 18). Similarly Patrick Brady defines fractals as "irregular shapes or number sequences that repeat themselves on varying scales" (Brady, 70).

<23> But once we look at the explanations of scientists, things are not quite as easy, and it is useful to return to the text which first tried to establish the fractal geometry of nature. At the beginning of chapter 18, Mandelbrot writes:

The bulk of this essay is devoted to fractals that are either fully invariant under similitudes or, at least, 'nearly' self-similar. As a result, the reader may have formed the impression that the notion of fractal is wedded to self-similarity. Such is emphatically not the case, but fractal geometry must begin by dealing with the fractal counterpart of straight lines ... call them 'linear fractals.' (Mandelbrot, 166 a, ellipsis in the original)
The familiar notion that fractal geometry is fully embedded in chaos theory needs to be qualified. The two fields overlap to some extent, but fractals are not necessarily self-similar, nor are they always irregular, and they also lead a happy life outside of chaos or complex dynamics. In particular many of the fractals discussed by Mandelbrot in the first chapters of his book are anything but chaotic -- the Koch Snowflake is a good example. Moreover, Mandelbrot points out that the self-similar Koch curve does not make a useful model for coastlines, because "[i]ts parts are identical to each other" (Mandelbrot 200 b) [11]. It should not be forgotten that Mandelbrot is primarily a mathematician, and chaos theory is only one aspect in his work. Thus many of the historical examples he offers are hardly concerned with chaos. The quote from Harvey (see above) about blood circulation quite obviously tried to establish an orderly sequence from above to below -- moreover, it was not particularly correct, and the microcosm-macrocosm correspondence cannot be revived in the light of recent mathematics and chaos theory.

<24> This, of course, strikes at the heart of quite a lot that has been written about complex dynamics, and especially of a multitude of texts on chaos theory and its anticipation in literature. Almost every text on the subject I have come across, and there are quite a few, takes fractals and self-similarity to be indistinguishable from chaos theory. Moreover, not only is self-similarity seen as an inherent feature of chaotic systems or processes, but frequently self-similar structures are even taken as a sufficient indication for chaos. But as self-similarity has been shown to be a common feature in concepts of the strictest order, it would make far more sense to see the various cases in art and literature as firmly embedded in and contributing to their respective historical paradigms. To put it quite rudely: many of the literary critics who hailed nonlinear chaos theory as the coming of a postmodern science and present their analyses as the final proof that literature paved its way may have focused on the non-chaotic aspects of fractals. I want to offer only two examples -- and again it can be safely assumed that anyone working in this context will be able to supply a few more. In her book Strange Attractors. Literature, Culture and Chaos Theory, Harriett Hawkins writes:

[T]he recognition of self-similarities in the nonlinear structures of nature itself, from cloud formations to coastlines, gives us a new way of perceiving the same phenomena (including Shakespeare's meta-dramatic self-similarities) obviously observable in past and present works of art. (Hawkins, 83)
But the passage she offers from one of Prospero's speeches in The Tempest (vi, i, 148-158), in which she detects a self-similarity that extends from the cosmic scale to the gorgeous palaces, to the theatre and finally to ourselves (cf. Hawkins, 83, actually, I fail to see this aspect in the speech), would be an evocation of the correspondence between macrocosm and microcosm. And the repetition of the number three throughout The Tempest that Hawkins presents as another example of its self-similarity (cf. Hawkins, 82) was formerly called 'parallelism.'

<25> My second example is from Peter Stoicheff's essay "The Chaos of Metafiction." In the context of the labyrinth of mirrors in John Barth's short story "Lost in the Funhouse" he states: "Mandelbrot's studies of chaotic systems revealed just such patterns, beautifully embodied in fractals, which 'above all ... meant self-similar'" (Stoicheff, 89, the quote is, of course, from Gleick, 103). But the strict self-similarity that is created by two opposing mirrors is not really chaotic, and neither are the Russian dolls that Stoicheff evokes on the next page. It is true, he quotes them from Gleick, but it may prove to be a serious mistake to accept all the examples and allusions we find in this text without any reservations. After all, Gleick tries to make his topic as palatable to his audience as possible, and simplifying images serve him well occasionally. But they reappear in literary criticism with the force of a scientific truth that is frequently challenged by the same texts.

<26> The problem we face here is that chaotic systems have some defining characteristics, and self-similarity is among them. It is the feature that sells, as it can easily be understood even by non-scientists -- possibly because it has a long pedigree in our culture as a whole. Literary criticism has hit upon that feature, but not on others like the Lyapunov exponent, and one of the reasons may be that, under the name of synecdoche, it has been an aesthetic principle in language and literature for a long time. But then the necessary qualifications were lost on the way, and self-similarity in the arts was regularly identified with a true vision of nature in general and chaos theory in particular. The result is a form of chaos in the field of 'literature and science.' Again this chaos is frequently based on repetitive patterns on various scales, but it is not quite the chaos we set out to explore.

References

Aristotle. Physics. http://classics.mit.edu/Aristotle/physics.3.iii.html.

Barth, John. Lost in the Funhouse [1968]. New York et al.: Bantam, 1978.

---. Tidewater Tales [1987]. London: Methuen, 1988.

Bohm, David. Wholeness and Implicate Order [1980]. London und New York: Routledge. 1995.

Bonheim, Helmut. "The Nature/Culture Dyad and Chaos Theory." Das Natur/Kultur Paradigma in der englischsprachigen Literatur des 19. und 20. Jahrhunderts (Festschrift zum 60. Geburtstag von Paul Goetsch). Ed. K. Groß. Tübingen: Narr. 1994, 8-22.

Bohr, Nils, "Nobel Prize in Physics Award Address" [1922], repr. in The World of Physics vol. 2. Ed. J.H. Weaver. New York: Simon and Schuster. 1987, 315-338.

Book of Kells [ca. 800], Luzern: Faksimile Verlag Luzern (www.faksimile.ch), 1990

Brady, Patrick. "Chaos Theory, Control Theory, and Literary Theory or: A Story of Three Butterflies." Modern Language Studies 20, 1990, 65-79.

Byatt, Antonia S. "Morpho Eugenia." Angels & Insects [1992], New York: Vintage, 1994.

Campbell, Joseph. The Hero With a Thousand Faces. New York: Bollingen Series, 1949.

Cavendish, Margaret. Paper Bodies. A Margaret Cavendish Reader. Ed. S. Bowerbank and S. Mendelson. Peterborough, Ontario: Broadview Press, 2000.

Chandler, Raymond. The Big Sleep [1939]. Stories and Early Novels, New York, Library of America, 1995.

Davidson, Donald. "What Metaphors Mean." On Metaphor. Ed. Sheldon Sacks. Chicago: University of Chicago Press, 1978, 29-45.

Davies, Paul. The Cosmic Blueprint. New York et al.: Simon and Schuster. 1988.

Davies, P.C.W. und Brown, J.R. The Ghost in the Atom. Cambridge: CUP. 1986.

Dawkins, Richard. The Selfish Gene [1976]. Rev. ed. Oxford and New York: Oxford UP, 1989.

Enuma Elish. Transl. L.W. King. http://www.sacred-texts.com/ane/enuma.htm.

Franzen, Jonathan. The Corrections. London: Fourth Estate, 2001.

Freud, Sigmund. Totem und Tabu [1913]. Frankfurt: Fischer, 1987.

Gebelein, Helmut. Alchemie [1991]. München: Hugendubel, 2000.

Gleick, James. Chaos: Making a New Science, London: Heinemann, 1988.

Gould, Stephen Jay. Ontogeny and Phylogeny. Cambridge, MA., and London, 1977.

Gross, Alan G., The Rhetoric of Science. Cambridge, MA, and London: Harvard University Press, 1990.

Hawkins, Harriet. Strange Attractors. Literature, Culture and Chaos Theory. New York et al.: Prentice Hall/Harvester Wheatsheaf, 1995.

Hayles, N. Katherine. "Designs on the body: Norbert Wiener, cybernetics, and the play of metaphor." History of the Human Sciences 3:2., 1990, 211-228.

Hobbes, Thomas. Leviathan [1651]. Cambridge: CUP, 1996.

Hofstadter, Douglas R. Gödel, Escher, Bach: an Eternal Golden Braid. New York: Basic, 1979.

Hölldobler, Bert und Wilson, Edward O. The Ants, Berlin et al.: Springer, 1990.

Johnson, Steven. Emergence: The Connected Lives of Ants, Brains, Cities, and Software. New York: Scrivener. 2001.

Joyce, David E. "Julia and Mandelbrot Sets." http://aleph0.clarku.edu/~djoyce/julia/julia.html.

Joyce, James. Finnegans Wake [1939]. London: Penguin, 1975.

Kuhn, Thomas S. The Structure of Scientific Revolutions [1962]. Second rev. ed. Chicago: University of Chicago Press, 1970.

Lacan, Jacques. "Le stade du miroir comme formateur des la fonction du Je." Écrits. Paris: Éditions du Seuil, 1966, 93-100.

Lakoff, George and Mark Johnson. Metaphors We Live By. Chicago and London: University of Chicago Press. 1981.

Lem, Stanislaw. Fiasco [1987]. Transl. Michael Kandel, New York et al.: Harvest, 1988.

Maclagan, David. Creation Myths, London: Thames and Hudson, 1977.Mandelbrot, Benoit, B. The Fractal Geometry of Nature [1977]. Rev. ed. New York: Freeman and Company, 1983.

Mead, Margaret. Growing up in New Guinea [1930]. London et al.: Penguin, 1942.

Miller, Arthur I. "Erotica, Aesthetics and Schrödinger's Wave Equation." It Must Be Beautiful: Great Equations of Modern Science. Ed. Graham Farmelo. London & New York: Granta, 2002, 110-131.

Nietzsche, Friedrich. "Über Wahrheit und Lüge im außermoralischen Sinne" [1873]. Unzeitgemäße Betrachtungen. München: Goldmann, 1964, 373-386.

Nikolaus Cusanus. De docta ignorantia, vol. II [1440]. Hamburg: Meiner. 1970; English translation Jasper Hopkins, http://www.cla.umn.edu/jhopkins/.

Otis, Laura. "The Other End of the Wire: Uncertainties of Organic and Telegraphic Communication." Configurations 9:2, 2001.

Poetic Edda. Transl. Henry Adams Bellows. http://www.sacred-texts.com/neu/poe/.

Pynchon, Thomas. Vineland, Boston et al.: Little Brown, 1990.

Rabelais, François. Gargantua and Pantagruel. Transl. J.M. Cohen. Harmondsworth: Penguin, 1985.

Santillana, Giorgio de and Hertha von Dechend. Hamlet's Mill. An Essay Investigating the Origins of Human Knowledge and its Transmission through Myth [1969]. Boston: Nonpareil, 1999.

Stoicheff, Peter. "The Chaos of Metafiction." Chaos and Order: Complex Dynamics in Literature and Science.Ed. K. N. Hayles. Chicago and London: University of Chicago Press, 1991, 85-99.

Swift, Jonathan. "On Poetry: A Rhapsody." Gulliver's Travels and Selected Writings in Prose and Verse. Ed. J. Hayward. New York: Random House, 1939, 790-803.

Thomas, Keith. Religion and the Decline of Magic [1971], London: Penguin, 1991.

Vanderbeke, Dirk. "Pynchons V oder: Wie man einen Buchstaben erzählt." Strukturen Erzählen, Ed. H.J. Wimmer. Wien: Edition Praesens, 1996, 516-541.

Vico, Giambattista. The New Science of Giambattista Vico (La Scienza nouva sconda, transl. Thomas Goddard Bergin und Max Harold Fisch). Ithaca und London: Cornell UP. 1984.

Wiener, Norbert. Cybernetics, or Control and Communication in the Animal and the Machine, 2nd ed. Cambridge Mass: MIT Press 1961.

Wilson, Edward O. Sociobiology: The New Synthesis (25th Anniversary Edition). Cambridge, MA, and London: Belknap Press. 2000.

Notes

[1] "A fractal is by definition a set for which the Hausdorff Besicovitch dimension strictly exceeds the topological dimension" (Mandelbrot, 15 b, italics in the original). [^]

[2] When a shorter version of this essay was presented at the Second European Conference of the International Society for Literature and Science in Århus 2002, Noëlle Batt pointed out that the term synecdoche has had a fixed meaning since antiquity and thus the transfer to the phenomenon of a structural self-similarity in language and literature seemed questionable. Instead, she suggested the mise-en-abîme as an analogy to self-similarity in other fields of knowledge. Her point is certainly valid. Nevertheless, I would like to maintain the term synecdoche. The tropes, far more than any of the other literary devices, structure our way of thinking, and thus I see the mise-en-abîme rather as one of the related sub-phenomena. But ultimately it is not the term but the principle that is of relevance here. [^]

[3] "[E]very actually existing thing contracts all things, so that they are, actually, that which it is" -- "omnis res actu existens contrahit universa, ut sint actu id quod est." (Nikolaus Cusanus, fol. 118) [^]

[4] "If you pay close attention to what has already been said, you will not have trouble seeing -- perhaps more deeply than Anaxagoras -- the basis of the Anaxagorean truth 'Each thing is in each thing'" -- "Si acute iam dicta attendis, non erit tibi difficile videre veritatis illius Anaxagorici 'quodlibet esse in quodlibet' fundamentum fortassis altius Anaxagora." (Nikolaus Cusanus, fol. 117). [^]

[5] "One aspect of the growth of personality is the process of individuation, of which the followers of Jung speak in cosmic terms: the separation of consciousness from unconsciousness, of 'ego' from 'non-ego,' being a psychic reproduction of such cosmogonic scenes as the separation of light from darkness or the parting of Father Sky from Mother Earth" (Mclagan, 9). [^]

[6] Cf. also Hayles, 223, for this aspect in Wiener's text. Hayles also mentions some of the following examples, but self-similarity is not an issue in her essay. [^]

[7] "For by art is created that great Leviathan called a Commonwealth, or State (in Latin, Civitas), which is but an artificial man, though of greater stature and strength than the natural, for whose protection and defence it was intended; and in which the sovereignty is an artificial soul, as giving life and motion to the whole body; the magistrates and other officers of judicature and execution, artificial joints; reward and punishment (by which fastened to the seat of the sovereignty, every joint and member is moved to perform his duty) are the nerves, that do the same in the body natural; the wealth and riches of all the particular members are the strength; salus populi (the people's safety) its business; counsellors, by whom all things needful for it to know are suggested unto it, are the memory; equity and laws, an artificial reason and will; concord, health; sedition, sickness; and civil war, death" (Hobbes, Leviathan, Introduction). Later in the text, Hobbes discusses the parts of a Commonwealth as " systems, which resemble the similar parts or muscles of a body natural" (Hobbes, ch. XXII). [^]

[8] In Ontogeny and Phylogeny, Stephen Jay Gould writes about Haeckel's law of recapitulation: "The idea of a relationship between ontogeny and the history of life was not the invention of a nineteenth-century German evolutionary zealot. Aristotle defended an analogical relationship between human development and organic history. The notion of a parallel between stages of ontogeny and sequences of adults [...] has been ubiquitous in biological theory" (Gould, 7). Gould also points out that the theory of recapitulation was abandoned in biology (cf. ibid. passim). Moreover, his book also contains a discussion of the primitive-as-child hypothesis, which he rejects as a "species of biological determinism" (Gould, 165). As early as 1930, the ethnologist Margaret Mead did research on animistic thinking in children and its assumed biological background when she studied childhood in New Guinea. She came to the conclusion that "[t]he results of this research were negative, that is, evidence was found to support the view that animism is not a spontaneous aspect of child thinking nor does it spring from any type of thought characteristic of immature mental development; its presence or absence in the thought of children is dependent upon cultural factors, language, folk lore, adult attitudes, etc., and these cultural forces have their origin in the thought of individual adults, not in the misconceptions of the children" (Mead, 217). [^]

[9] Book of Kells, fol. 34r: Chi Rho-page. [^]

[10] In consequence, there are totalizing statements like Alan Gross's claim that "science is full of metaphor, and it is the nature of metaphor deliberately to misname" (Gross, 80). This, of course, only repeats Nietzsche's well-known dictum that metaphors are just a kind of lies (cf. Nietzsche, passim) and ignores some decades of productive work on the cognitive function of metaphor in literature, science, and language in general (cf. for example Davidson, 41). [^]

[11] The physicist Paul Davies is equally cautious in his approach. In 1988, that is 11 years after the publication of Mandelbrot's The Fractal Geometry of Nature, he writes about fractals: "The question is, are they of interest only for mathematicians, or are there fractals in nature? The Koch curve is only meant to be a crude model for a coastline, and further processing and refinement is necessary before realistic coastal shapes are generated. Nevertheless, it could be argued that the approximation of a coastline to a fractal is better than its approximation to a smooth curve, and so fractals provide a more natural starting point for the modelling of such forms" (Davies, 60). [^]