Romantic Chaos: The Dynamic Paradigm in Novalis's Heinrich von Ofterdingen and Contemporary Science

by Joyce S. Walker
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Romantic Chaos: The Dynamic Paradigm in Novalis's Heinrich von Ofterdingen and Contemporary Science
Author:
Joyce S. Walker
Year: 
1993
Publication: 
The German Quarterly
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66
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1
Start Page: 
43
End Page: 
59
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English
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Abstract:

S. WALKER University of California, Davis

Romantic Chaos:
The Dynamic Paradigm in Novalis's Heinrich von
Ofierdingen and Contemporary Science

The idea of chaos pervades Novalis's novel Heinrich von Ofterdingen and plays a crucial part in his Romantic vision, providing the basis for both natural and poetic creativity. Chaos in Novalis is a dynamic paradiem which challenges the very presuppositions of Enlightenment science and is a provocative anticipation of modernity. Indeed, it was not until a cen- tury and a half after the novel's publication in 1802 that acorrespondingtrendemerged in science itself. Like Novalis, 20th-century chaos theorists overturned the Newtonian model of a predictable, orderly universe in their quest to describe the disorder apparent in nature. In an attempt to explicate chaos, both as a concern in early German Romanticism and later, in contemporary scientific thought, I shall first examine chaos in Novalis's novel, and then, in a gradual unfolding of the complexities of scientific chaos theory, explore the shared features of this dynamic paradigm. Accord- ingly, this discussion begins with chaos theory's apprehension of nature's diversity and unpredictability and its preference for nonlinear, aperiodic, and organic models. Next, the concept of "sensitivity to initial conditions"isexp1icated and compared with Novalis's concept of Zufall, followed by an examination of phase transitions in chaos theory and in Heinrich von Ofterdingen vis- Q-vis their mutual reliance on analogy and metaphor. Then, the concept of "scaling" in chaos theory, the use of fractals to describe the geometry of nature, and the centrality of the principle of self-similarity are all in- troduced and subsequently applied to the content and structure of Novalis's novel. Finally, the interrelationship of the chaos paradigm in Novalis and in contemporary science leads to some reflections on the Romanticism of science and the autonomy of poetry.

History in Heinrich uon Ofterdingen is bracketed by a Golden Age, regarded with nostalgia and anticipation, of which chaos is the harbinger. In order fully to grasp Novalis's idea of chaos, it is important to examine some details of this context. Early in the novel, the merchants tell Heinrich about the mythical, harmonious past: 'ln alten Zeiten, md3 die ganze Natur leben- diger und sinnvoller gewesen seyn, als heut zu Tage" (1:210). This theme is repeated in their tale about Atlantis, in which the young poet sings 'ton der uralten, goldenen Zeit" and its return (1: 225). Klingsohr's "Mar- chen" brings the Golden Age into sharp focus, as Fabel sings at the spinning wheel of her vision of harmony and unity (1: 302- 03). Finally, it is Fabel (Poesie) herself who announces the return of the Golden Age at the 'Marchen's" end, "Die alten Zeiten keh- ren zuriick" (1: 310), culminating in a description of the harmony and fertility of this new world. At the beginning of the novel's unfinished second half, Astralis an- nounces the advent of the Golden Age, where there is "Keine Ordnung mehr nach Raum und Zeit / Hier Zukunft in der Ver- gangenheit" (1: 3 18). Tieck's "Bericht" sug- gests that the finalclimaxofthe novel would have been the inauguration of the Golden

The German Qz~arterly 66.1(Winter1993) 43

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Age (just as it was in Klingsohr's "Mar- chen") symbolized by a child: "Dieses Kind ist die Urwelt, die goldne Zeit am Ende" (1: 368).

In this context, chaos is creative poten- tial. It provides the necessary conditions for the advent of the Golden Age because it breaks up the static and stultifyingorder of history, and its flux gives rise to a new and dynamic order. In Klingsohr's "Marchen," the Phoenix, arising from the ashes of ruin, incarnates this mythology of rebirth and is an embodiment of Fabel's statement: "Die Asche meiner Pflegemutter mu13 ich sam- meln, und der alte Trager mu13 wieder aufstehn, da13 die Erde wieder schwebe und nicht auf dem Chaos liege" (1:310). Playing a symbolic part in Klingsohr's "Marchen," chaos and its agent, war, appear in a vision in the treasure chamber of the Moon King. Later, they are not permanently destroyed, but are, instead, magically banished into a chess set after the advent of the Golden Age, "Aller Krieg ist auf diese Platte und diese Fipren gebannt" (1: 314), thus suggesting the necessity of their continued, although constrained, presence. In the novel's un- finished second half, Heinrich was to have experienced chaos and war firsthand prior to his final transfiguration and the arrival of a Golden Age of unity, harmony, and transcendence.

Just as chaos is essential for the advent

of the GoldenAge, so, too, must it play a role

in the poet's unifying vision, which moves

beyond ordering reason and encompasses

all of life's antinomies. Klingsohr advises

Heinrich that "das Chaos mu13 in jeder

Dichtung durch den regelmaaigen Flor der

Ordnung schimmern" (1: 286). The multi-

plicity of outer and inner life, or nature and

human nature, which is the physical

manifestation of chaos, may be melded into

unity only by the poet who is conscious of

the disorder which underlies creation. That

Novalis's poetic understanding ofchaos was

powerfully shaped by his scientific training

has been convincingly argued by Peter

Kapitza, John Neubauer, and Dennis

MahoneY1 Kapitza roots Novalis's concept of chaos in the 18th-century understanding of chemistry as both "Scheidekunst" and "Mischungskunde" (14). Hence, as John Neubauer contends, Novalis's chemical metaphors emphasize not only mixing and amalgamation but also dissolution, thereby distinguishing his imagery from the quest for orgarlic synthesis characteristic of his fellow Romantics (130) and assuring the centrality of chaos in his poetics, where it is the agent of both synthesis and dissolu- tion. Novalis wrote:

Vor der Abstrakzion ist alles eins, aber eins wie Chaos; nach der Abstrakzion ist wieder alles vereinigt, aber diese Vereini- gung ist eine freye Verbindung selbst- sthndiger, selbstbestimmter Wesen-Aus einem Haufen ist eine Gesellschaft gewor- den-Das Chaos ist in eine mannichfaltige Welt verwandelt. (2:455-56)

Thus, inNovalis'spetics, chaosdestroys and recreates new unities on all levels of ex- perience, extending the chemical metaphor to the biological sphere and even (as the pre- vious aphorism shows) to the sociopolitical realm. This prompts Neubauer to argue that for Novalis nature is a fluid, rather than fured, construct, "a multiplicity of hypotheti- cal alternatives existing side-by-side and changing in time" (135). So, when Novalis applies this metaphor to the political realm, he writes of the essential role played by those harbingers of chaos: war and revolution. War, because it literally and metaphorically dissolves old "solutions," creates, to borrow Neubauer's phrase, a new "multiplicity of hypothetical alternatives." Novalis expresses this through the symbolic presence of war in Klingsohr's "Marchen." When Klingsohr teaches Heinrich that "im Kriege . . . regt sich das Urgewasser" (1:285), he applies to the political realm language which suggests an analogy with chemistry. There- fore, it comes as no surprise that Novalis describes chemical reactions and cataclys- mic natural processes when he addresses the issue of revolution in "Glauben und Liebe":

Michtige Therschwemmungen, Verande-
rungen der Klimate, Schwankungen des
Schwerpunkts, allgemeine Tendenz zum
ZerflieRen, sonderbare Meteore sind die
Syrnptome dieser heftigen Incitation, de-
ren Folge den Inhalt eines neuen Welt-
alters ausmachen wird. So nijthig es
vielleicht ist, da13 in gewissen Perioden
alles in FluR gebracht wird, um neue,
nothwendige Mischungen hervorzubrin-
gen, und eine neue, reinere Kristallisation
zu veranlassen, so unentbehrlich ist es
jedoch ebenfalls diese Krisis zu mildern
und die totale ZerflieRung zu behindern,
damit ein Stock iibrig bleibe, ein Kern, an
den die neue Masse anschieRe, und in
neuen schonen Formen sich um ihn her
bilde. (2:490)

This necessary bringing-of-everything-intoflow in order to create new forms is the highest taskof the pet, whosevision not only encompasses the "Lichter, Farben und Gestalten" of the natural world, but is also trained inward toward the magical and mysterious "Geist des Lebens" (1: 209,259). This is the vision of the young poet in the Atlantis tale in Heinriclz von Ofterdingen, whose vision/visage ("Sein Blick schien trunken in eine geheimere Welt hiniiber zu schauen"[l: 2251) echoes the visiodvisage of Klingsohr, whose "edles Ansehen" and "durchdringende und feste Augen" (1: 270) are evidence of this same transformation of vision from outer multiplicity to inner unity. This flowing together of inner and outer vision, perception and object, real and ideal, the quotidian and the wondrous, accom- plishes the poetic taskexpressed by htralis: "Und was vordem allttiglich war / Scheint jetzo fremd und wunderbar" (1: 318).

Novalis's understanding of the role of chaos in natural and artistic creation rep- resents a shift away from the dominant 18th-century w~rldview.~The

predominant scientific theories, like those of Newton, were based on a methodology which in- volved scouring nature's multiplicity for regularities and principles, identifying "1aws"and thus making possible the predic- tion and control of natural forces. With its

faith in the capacity of the human mind to accomplish this vast systematization of the natural world, Enlightenment science rare- ly explored phenomena which were disor- dered, random, or singular. It was not until the mid-20th century when chaos theory, a new scientific paradigm, began to explore those boundaries of science where order crossed the threshold into disorder-into the previously indescribable realm of chaos. James Gleick, in Clzuos: Making a New Science, traces the emergence of the chaos paradigm across many disciplines, includ- ing mathematics, physics, and biology. He repeatedly emphasizes the fact that chaos theory could have been discovered long before, but that rationalistic science lacked any incentive: 'The stable solutions were the interesting ones. Order was its own reward" (65).3 The vision of scientists was therefore faulty. They looked at the world and, like the Schreiber in Klingsohr's "Marchen," they saw only quantifiable and orderly phenomena which reason could subjugate to human ends, and they ignored random and useless "noise." The insuf- ficiency of this approach is epitomized in the scene in which the Schreiber examines a magnetized metal splinter: "Der Schrei- ber besah es und drehte es mit vieler Leb- haftigkeit herum, und brachte bald heraus, dd es sich von selbst, in der Mitte an einem Faden aufgehangt, nach Norden drehe" (1: 294). For Ginnistan (fantasy), the splinter is transformed into a snake which bites its tail, an archetypal symbol for both eternity and sexual union (Pfotenhauer 244). An- noyed, the Schreiber returns to his pen and paper and diligently sets down his observa- tions about the utility of his discovery: "Er schrieb alles genau auf, und war sehr weitlaufig iiber den Nutzen, den dieser Fund gewahren konne" (1: 295). However, when his report is dipped in the water of wisdom, it emerges completely blank, thus heralding the new paradigm suggested in Novalis's poem, which begins with a rejec- tion of utilitarian and reductionist vision: 'Wenn nicht mehr Zahlen und Figuren /

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Sind Schlussel aller Kreaturen. .."(I:360).

Novalis elsewhere criticizes this faulty vision, disparaging those who "sich gewoh- nen wurden alles GroRe und Wunder- wiirdige zuverachten, und als todte Gesetz- wirkung zu betrachten" (3: 50E49). In Gleick's description of how an important chaos theorist broke these rules of vision and realized that he was "seeing things that physicists had learned not to see" (45), there are provocative reminders of Novalis, for whom the poetic vision perceives and incor- porates chaos. Moreover, against all expec- tations, scientists soon observed that chaotic phenomena appeared in unexpected places (often within otherwise order- ly systems) and that chaos possessed an order all its own. Gleick notes: "Simple deterministic models could produce what looked like random behavior. The behavior actually had an exquisite fine structure, yet any piece of it seemed indistinguishable from noise" (79). Gradually, it became ap- parent to researchers that chaotic behavior could be most often observed in systems which were nonlinear and aperiodic.

"Linear" and "nonlinear" describe two types of mathematical equations. Linear mathematics may be imagined as contain- ing all those equations whose solutions result in the graphing of a straight line, whereas nonlinear equations may be graphed as circles, parabolas, sine waves, and other n~nlinear shapes. It is difficult to solve these equations; therefore, limited solutions are often approximated by resort- ing to linear equations. However, chaos theory is primarily concerned with employ- ing nonlinear mathematics, for "Nature is . . . [nonlinear] in its soul" (Gleick 68).Chaos theorists assert that the linear model is inappropriately applied to nature's essen- tial nonlinearity and that exclusive applica- tion of the linear model does violence to the earth:

It may not be a deep truth toassert that our

world is nonlinear and complex; everyday

experience has never taught us otherwise.

Yet physics and mathematics, and other sciences following them, have managed to ignore the obvious. Focussing on simple problems that they could solve, these sciences had a strong impact on technology and thereby drastically changed the sur- face of our planet. It is now being felt, how- ever, that more is needed than knowledge of the linear phenomena. (Peitgen and Richter 20)

Nature's nonlinear systems are capable of a self-regulation that is impossible in linear systems. A linear system, given a "shove" by an external factor, will change its trajectory and cannot right itself, much the way a bil- liard ball alters its course when struck by another. A nonlinear system, on the other hand, has the ability eventually to correct itself and return to its own starting point. The human body is a model of such a non- linear system, as exemplified in the ability of the body to recover from illness or injury (Gleick 292).

In addition to this first requirement of nonlinearity, chaotic systems must also pos- sess the quality of aperiodicity. Periodic sys- tems repeat or reproduce themselves without variation, over and over again, and therefore are predictable. From the predict- ability of the pendulum swing and the determinism of physics to the postulation of a Clockmaker deity, Enlightenment science expressed confidence that anunder- standing of the orderly laws of the universe would result in the accurate ability to predict future phenomena (Gleick 12, 40). However, despite impressive efforts, cer- tain aperiodic aspects of nature, such as weather, continue to defy prediction. Ex- pressed as mathematical equations and portrayed on a graph, an aperiodic system's cycles show a distinctly repetitive pattern, but never an exact replication. Like snowflakes, these systems manifest pat- terns which are similar, but never complete- ly predictable. This is the context of Gleick's remarks that "Chaos cuts away at the tenets ofNewton's physics"and that "Chaos poses problems that defy accepted ways of

WALKER:

working in science" (5, 6). Moreover, Gleick's following assessment of chaos theory's success in describing vital and un- predictable natural processes recalls Novalis's Romantic disillusionment with Enlightenment science and his exaltation of natural process and flow:

Where Chaos begins, classical science stops. For as long as the world has had physicists inquiring into the laws of na- ture, it has suffered a special ignorance about disorder in the atmosphere, in the turbulent sea, in the fluctuations of wild- life populations, in the oscillations of the heart and brain. The irregular side of nature, the discontinuous and erratic side- these have been puzzles to science, or worse, monstrosities. (3)

Indeed, Novalis'scausticcomment on the (for him) failed program of Enlightenment science is a tribute to this unpredictability, irregularity, and irreducibility of nature: "Schade dalj die Natur so wunderbar und unbegreiflich, so poetisch und unendlich blieb, allen Bemiihungen sie zu modernisieren zum Trotz" (2: 516). In this para- digm, Enlightenment science must fail be- cause it is an unpardonable static reduction of the multifaceted, dynamic whole. Chaos theory, on the other hand, has been heralded as the end of the compartmentalizing, reduc- tionist program in science and as a quest to unveil nature in all its wholeness (Gleick 304).~Interestingly, chaos theory appeals to our visual perception of nature's multiplicity to argue its case. Benoit Mandelbrot, inven- tor of the celebrated Mandelbrot set, ex- presses his agreement with the ideas of mathematician Jean Perrin concerning the contrast between the reductionist stance of Enlightenment science and the eye's percep tion of the world:

L'id6e classique est bien certainement que l'on peut dkcomposer un objet quelconque en petites parties pratiquement homo- &nes. . . . Loin que cette conception soit impos6e par l'exp&ience, j'oserd presque dire qu'elle lui correspond rarement. Mon ~ilcherche en vain une petite region "pratiquement homog8ne1' sur ma main, sur la table oh j'6cris, sur les arbres ou sur le soleil que j'apercois de ma fen6tre. 5 (6)

Here, the mediated vision of Enlightenment science, as the "dissecting" eye aided by microscopes and telescopes, is replaced by the unmediated vision of the eye's direct ob- servation of the rich diversity of nature.

Thus, aschaos theorists focused on non- linear and aperiodic natural systems such as weather, their laments over the inade- quacy of numerical descriptions sound at times like a critique of the Schreiber in Heinrich von Ofterdingen,, who bedecks himself with the numbers and geometrical shapes ("Zahlen und Figuren") which spring forth when he is coincidentally 'bap- tized" with the water of wisdom (1:294). Gleick writes that scientists like Edward Lorenz began by wondering whether "weather science had a flavor that could not be expressed by talking about averages" and statistics, and ended up making sig- nificant contributions to the understanding of the chaos inherent in natural systems (12). The mathematician Mitchell Feigen- baum, another important contributor to the new science, expressed to Gleick his mistrust of mathematical adequacy:

I truly do want to know how to describe clouds. But to say there's a piece over here with that much density .. . to accumulate that much detailed information, I think is wrong. It's certainly not how a human being perceives those things, and it's not how an artist perceives them. Somewhere the business of writing down partial dif- ferential equations is not to have done work on the problem. (187)

Here, the preference of chaos science for the vital, turbulent, organic processes of non- linear and aperiodic natural systems such as weather is suggestive of the prevalence of organic metaphor in early German Roman- ticism as epitomized by Novalis. As we will see later, Heinrich uon Ofterdingen is packed with dynamic, organic metaphors.

First, however, it is important to round

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out the description of chaos theory with a discussion of the key concept of sensitivity to initial conditions. Enlightenment sci- ence, with its focus on linearity and predict- ability, was predicated on a concept of natural law which may be illustrated by returning to the analogy of the billiard table. That is, calculating the force and angle of the strike of the cue, it is possible to predict the trajectory of the ball. Any al- teration in the force or angle of the cue will result in a predictably different outcome. Gleick summarizes:

Scientists marching under Newton's ban- ner actually waved another flag that said something like this: Given an approximate knowledge of a system's initial conditions and an understanding of natural law, one cancalculate the approximate behavior of the system. This assumption lay at the philosophical heart of science. (15)

Chaos theory challenged this assumption through the discovery that nonlinear and aperiodic systems are tremendously sensi- tive to initial conditions and that a small al- teration in these conditions could produce unpredictable and amazing variations. This realization also undermined another funda- mental assumption of Newtonian physic- namely, that simple causes lead to simple results, and complex causes to complex results. Instead, chaos means that complexity may in fact have simple causes, at- tributable to minute changes in initial con- ditions. In Gleick's words:

The modern study of chaos began with the creeping realization in the 1960's that quite simple mathematical equations could model systems every bit as violent as a waterfall. Tiny differences in input could quickly become overwhelming differences in output-a phenomenon given the name "sensitive dependence on initial condi- tions." (8)

Perhaps the most familiar example of this phenomenon is the snowflake, produced by an essentially deterministic system but manifesting apparently infinite variety. The fact that no two snowflakes resemble each other is a result of the chaos introduced in the initial conditions of their formation. Minute variations in such factors as temperature, humidity, and wind result in this amazing multiplicity of design (Gleick 311).

This scientific understanding of sen- sitivity to initial conditions has its counter- part in Novalis's concept 0fZufal1.~ Gleick's description of the important role played by sensitivity to initial conditions in daily life could easily be read as a definition ofZufal1. Returning to the vision paradigm, he writes:

Physicists had learned not to see chaos. In daily life, the . . . quality of sensitive de- pendence on initial conditions lurks everywhere. Aman leaves the house in the morning thirty seconds late, a flowerpot misses his head by a few millimeters, and then he is run over by a truck. . . . Small perturbations in one's daily trajectory can have large consequences. (67)

This pervasive manifestation of chaos in the quotidian is embodied in Arctur, the king in Klingmhr's "Marchen" in Heinriclz von Ofterdingen, who represents the 'Gist des Lebens, der dem Menschen nur als 'Zufall' fal3bar ist" (Pfotenhauer 242). Elsewhere in the novel, Novalis asserts that the hero and the poet each shape Zufall in their own way. Of heroes, Novalis writes: "Alle Zufa'lle wer- den zu Geschichten unter ihrem Einfld, und ihr Leben ist eine ununterbrochene Kette merkwiirdiger und glanzender, ver- wickelter und seltsamer Ereignisse" (1: 266). Likewise, the poet observes that "Mannich- faltige Zufa'lle schienen sich zu seiner Bildung zu vereinigen" (1: 267-68). Zufall is central, for, according to Novalis, "Der Dich- ter betet den Zufall an," as an irruption of the transcendent, the mystical and mysteri- ous essence of life (3: 449). Manfred Dick writes:

Zufalligkeit bedeutet nicht ein selbstversGndliches, NChtweiter des Vorliegen, dessen Herkunft nicht erMCrt werden kann und iiber das sich auch weiter keine Gedanken lohnen. Sie

WALKER:

bedeutet urngekehrt, dal3 das vorliegende Wlrkliche ein Unerklarbares und Geheim- nisvolles wird, d& es in sich und durch sich in ein Unfdbares und Unbekanntes verweist. (398)

Thus, Zufall, as an intrusion of the magical and mysterious, presents humans with an opportunity for creativity and transcendence. WhenNovalis writes, "Alle Zufalle un- sers Lebens sind Materialien, aus denen wir machen konnen, was wir wollen" (2:436), he is emphasizing the creative and shaping na- ture of human freedom, despite the ap- parently contradictory and rather deter- ministic conclusion that Zufall is not arbi- trary but in fact has meaning as an expres- sion of the transcendent. This paradox is made explicit where he writes, "Auch der Zufall ist nicht unerg7undlich-er hat seine Regelmafiigkeit" (3: 414), and it provides another remarkable parallel to the new scientific understanding of chaos as a deter- ministic system which generates random- ness-randomness with its own underlying order (Gleick 250, 252). In fact, chaotic be- havior most often occurs in the transitions between and on the boundaries of systems, demonstrating remarkable similarities even in disparate systems. From nature's ap- parent disorder, there emerges a new fun- damental, underlying order. According to Gleick

The rolling of eddies, the unfurling of ferns, the creasing of mountain ranges, the hollowing of animal organs all followed one path. . . . It had nothing to do with any particular medium, or any particular kind of difference. The inequalities could be slow and fast, warm and cold, dense and tenuous, salt and fresh, viscous and fluid, acid and alkaline. At the boundary, life blossoms. (198)

Eventually, having started from the sen- sitivity of initial conditions as epitomized by the infinite variety of the snowflake, chaos theorists saw surprising universality in the paths (or mathematical forms) of chaos.7

To understand why this is possible, it is

important to remember that many non- chaotic, orderly systems become chaotic when certain limits are reached. One ex- ample of this is wildlife population biology, where graphs depicting fertility rates be- have chaotically when those rates exceed a certain limit (Gleick 69-77). Another ex- ample is the phenomenon of phase transi- tions. Phase transitions occur when sub- stances change their fundamental state, such as when water freezes to become ice, or is heated to become vapor. At these boundaries, substances and systems move from orderly to chaotic and then back to orderly again when the new state is achieved. Feigenbaum's discovery that all phase transitions follow the same "rules" (whether water is boiled, or metals are mag- netized) was at first met with incredulity, but the Feigenbaum number, as his equa- tion is now called, has conclusively demonstrated the universality of these phenomena (Gleick 160-61, 180). Again, a certain order emerged within cha~s.~

Physical and "spiritual" phase transi- tions pervade the structure and themes of Heinriclz von Ofterdingen. As discussed above, chaos itself is the transition to the Golden Age. Within Klingsohr's "Marchen," the Golden Age is heralded by the melting of the winter ice and the return of flow to the world. Likewise, metal undergoes a phase transition to magnetism, and the magnetized splinter from Eisen's sword guides Eros to the North, to his beloved Freya. Moreover, the setting of the novel in the Middle Ages is a further image of tran- sition, as Novalis explains in the novel:

In allen aerghgen scheint, wie in einem Zwischenreiche, eine hohere, geistliche Macht durchbrechen zu wollen; und wie auf der Oberflache unseres Wohnplatzes, die an unterirdischen und iiberirdischen Schatzen reichsten Gegenden in der Mitte zwischen den wilden, unwirthlichen Ur- gebirgen und den unermeRlichen Ebenen liegen, so hat sich auch zwischen den rohen Zeiten der Barbarey, und dem kunstreichen, vielwissenden und begiiter-

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ten Weltalter eine tiefsinnige und romantische Zeit niedergelassen, die unter schlichtem Kleide eine hohere Gestalt ver- birgt. (1:204)

This transition motif is also embodied in the theme of twilight, which runs throughout the novel and which Peter Kiipper examines in depth, remarking: "Kein Zufall daher, da13 computer models, the essential tools of chaos theory, are descriptions of reality which are essentially metaphorical and analogical. The analogical potential of mathematics was apparent to Novalis as well, for he wrote: 'Zahlen sind wie Zeichen und Worte, Erscheinungen, Reprasenta- tionen . . ." (3: 593). Gleick refers to 20th- century physicists, like John von Neu-

der 'Ofterdingen' in einer '~ber~angszeit' mann, who assert that 'The sciences do not

spielt, so wenig auch, da13 dem tageszeit- lichen ubergang, der 'Dammerung,' in diesem Roman eine bedeutsame Rolle zufallt" (84). Moreover, poetry itself is the means of transition in which reality is trans- formed into Mth-clzen.As in the first half of the novel, "Die Erwartung," reality is trans- formed into the ideal, culminating in the al- legory of Klingsohr's "Marchen," so in the second half, "Die Erfillung," ideal is trans- formed into reality. In the first half, vision of the outer world is transformed into inte- riority and disappears, as epitomized by Heinrich's exclamation, "Die reiche Land- schaft ist mir wie eine innere Fantasie" (1: 279). The second half would have concluded (accordingto Tieck's 'Bericht") with the com- plete merger of the ideal and real, exemplified by Astralis, the spirit created from Heinrich and Mathilde's kiss. In this ex- change, however, reality is actually taken over by the ideal, for it can't be described in ordinary, visual terms. Instead,Tieckreports Novalis's design: "Die Mahrchenwelt wird ganz sichtbar, die wirkliche Welt selbst wird wie ein MciJz~*clzenangesehn" (1: 368). These few examples of the pervasiveness of the transition motif in Heinrich uon Oflerdingen suggest that the novel is thematically and structurally dynamic. Its emphasis on tran- sitions--on crossing boundaries-coupled with the Romantic program which sought to spring over boundaries into the infinite, coin- cides in a remarkable way with these same concerns in scientific chaos theory.

The fact that science, like literature, relies heavily on analogy and metaphor is a fact which did not escape some chaos theoristsg Mathematical equations and try to explain, they hardly even try to inter- pret, they mainly make models" (273). In this view, then, science proceeds by analo- gizing-by transforming reality into ideas. Surely this sort of ideal activity finds an echo in Novalis, in Heinrich's aforecited ad- mission that "die reiche Landschaft ist mir wie eine innere Fantasie" (1: 279). More- over, in the absence of the corresponding idea, certain aspects of reality may be over- looked, as the abundant evidence of chaos in nature was overlooked by scientists for centuries for lack of a model. Paraphrasing Thomas S. Kuhn, Gleick writes: ". . . you don't see something until you have the right metaphor to let you perceive it"(262). Chaos theory provided the right metaphors.

Metaphor and analogy are the predom- inant devices employed in Heinriclz uon Oflerdingen.Yet the fact that so many of the novel's images are analogies of transition is reminiscent of Feigenbaum's discovery that all phase transitions follow the same rules -that a certain order emerges from chaos. In order to transform reality into idea, Novalis masterfully wields his "Zauberstab der Analogie" (3: 518). In Heinriclz uon Oflerdingen, reality becomes idea through the medium of the word-through the su- preme art of poetry. Even pictures are ex- plained as manifestations of the verbal: they are '%vie verkorperte Worte" (1:264). An examination of the axes of space and time in the novel reveals the complexity of Novalis's layers of organic metaphors. On the spatial axis, the near is identified with the present and the distant with the future and past; heightsbeaven represent the fu- ture and depthslearth the past; and North represents the future while South is the past (Kiipper 91-92). These spatial po- larities, representing time, are echoed on a time axis which accords with human stages of life: children are identified with the primordial paradise of the past and the Golden Age of the future, while age (em- bodied in Sylvester, the hermit, and the Bergmann) recalls the paradisiacal past and points the way to the future (Kupper 95). It is the word as poetry, personified in Fabel in Klingsohr's "Marchen," which ef- fects the union of polarities represented in these metaphors, and catalyzes the transi- tion to the Golden Age: 'Qurch die dichteri- sche Verwandlung der Welt wird alles wieder zu Poesie werden" (Mahl398).

Chaos science's study of the boundary regions between chaos and order led to the concept of scaling. The geometrical, or mathematical, convergence observed in phase transitions is evidence for this idea. Mandelbrot provocatively suggested that the length of the coastline of Britain is per- haps infinite. A measurement calculated from an aerial photograph will be shorter than a measurement made by someone on foot using a surveyor's tape. That measure- ment, in turn, would be dependent on the geometric angles at which the tape is placed. Indeed, at the microscopic level, taking into account all the eccentricities of cliffs and rocks, the coastline appears longer still. How long is it?

Mandelbrot's question, which can only be answered in relationship to the scale on which things are measured, led him to explore the concept of fractals. Fractals are geometric shapes which "exist" in fractional space (for instance, in 1% dimensions). They have the unique quality of containing infinity within afinite space, for their struc- ture consists of repetitions of the original shape, always onsmaller and smaller scales (see Fig. 3). Fractals are self-similar, and "self-similarity is symmetry across scale. It implies recursion, pattern inside of pattern" (Gleick 103). In fact, Mandelbrot believed that the shape of the coastline, as well as other jagged and fragmented irregularities of nature, was based on the fractal. The astonishing underlying order explained through this new model succeeded in de- scribing nature where previous models failed. Mandelbrot wrote:

Dans son effort mur dBcrire le monde, la science ne peut que procBder par des series d'images ou modhles de plus en plus "rBalistes." Les plus simples ont Bt6 des continus parfaitement homo&nes, tels un fil ou un cosmos de densit6 bien d6finie et uniforme, ou un fluide de temgrature, densit6, pression et vitesse Bgalement bien definies et uniformes. . . . Mais il est d'autres domaines, oh la realit6 dont on traite se rBvhle dtre irkgulihre h l'extrkme, put-dtre mdme h l'infini, h tel point que le modkle continu parfaitement homogkne de~oit. . .1°(3)

Analogies and metaphors in Heinriclz von Ofterdingenmaybe viewed as fractals which have relationships of scale to one another. Astralis expresses the essence of this model in his declaration, "Jedes in Allen dar sich stellt" (1: 319), as, in Klingsohr's "Marchen," Fabel answers the Sphinx, 'Wer kennt die Welt?-Wer sich selbst kennt"(1: 308). Thus, layers of organic metaphors, when peeled away, often reveal more layers on a different scale, as, for instance, in the series which begins with the metaphor of Mathilde as a flower ("Eine nach der aufgehenden Sonne geneigte Lilie war ihr Gesicht . . ." [I: 2711), which then moves to the metaphor of flowers as the "Ebenbilder der Kinder" (1: 329) and culminates in the metaphor of children as the incarnation of the Golden Age (1: 329-30, 368). Scaling also applies to social reality: 'Wie Heiligtumer wird eine weisere Nach- kommenschaft jede Nachricht, die von den Begebenheiten der Vergangenheit handelt, aufsuchen, und selbst das hben eines Ein- zelnen unbedeutenden Mannes wird ihr nicht gleichgiiltig sein, da gewil3 sich das groJ3e Leben seiner Zeitgenossenschaft darin mehr oder weniger spiegelf' (1: 258). Novalis sometimes calls this process "transubstan- tiation": "In unsern Zeiten haben sich wahre Wunder der Transsubstantiation ereignet. Verwandelt sich nicht ein Hof in eine Familie, ein Thron in ein Heiligthum, eine koniglicheVermiihlung in einenewigen Her- zensbund?" (2: 498).

Finally, Heinrich uon Ofterdingen re- sounds with the conviction that humancon- sciousness represents a finitely-bounded infinity. Novalis wrote: 'TJach Innen geht der geheimnifivolle Weg. In uns, oder nir- gends ist die Ewigkeit mit ihren Welten- die Vergangenheit und Zukunft" (2: 418). Ultimately, Novalis uses allegory as a device for the artistic portrayal of reality's fractal structure. Klingsohr's "Marchen" is a rendering of the novel's themes and metaphorson another scale-a fractal sum- mation of the tales, dreams, and action of the novel. And according to Tieck's "Be- richt," the novel was to have ended on yet another scale as an allegory, where "Alles fliefit in eine Allegorie zusammen" (1:369).

Pursuing the fractal principle of self- similarity, Mandelbrot created another "picture'' of reality, with stunning visual results. The Mandelbrot set (see Fig. 1and 2) is a computer-generated picture of the iterations of a nonlinear equation which seeks to define the boundary region repre- senting a transition between chaos and order (Peitgen and Richter 8). Each detail, when magnified, reveals yet another scale within its inner universe. Even magnified a million times, to the very limits of tech- nology, the original shape of the Mandelbrot set is still distinguishable within, signaling the recommencement of the entire series, and so on, into infinity, so that "every new detail was sure to be a universe of its own, diverse and entire" (Gleick 229). The team of scientists who developed the extraordi- nary set of photographs which made the set famous wrote:

Our pictures . . . deal with chaos and order, and with their competition or coexistence. They show the transition from one to another and how magnificently complex the transitional region generally is. . . . The processes chosen here come from various mathematical or physical prob- lems. They all have in common the com- petition of several centers for domination bf a plane. Asimple boundary between ter- ritories is seldom the result of this contest. Rather there is unending filigreed entanglement and unceasing bargaining for even the smallest areas. (Peitgen and Richter 4)

Perhaps the most visually striking detail about the set is the similarity of its shapes to those of nature: bifurcations (branching structures characteristic of chaos) suggest ferns, river drainage basins with branching tributaries, or the human circulatory sys- tem." However, it is important to remember that the set itself is a metaphor, rather than replication, of reality. The creators of the photographs cautioned: 'The infinite micro- scopic depth to which the self-similarity seems to reach is a mathematical construct that does not exist in the real world. Physical objects are seldom self-similar over more than four orders of magnitude" (Peitgen and Richter 18). But it is a metaphor which

recalls Novalis's vision in Heinriclt uon Ofter- dingen, in which "Das Weltall zefallt in un- endliche, irnmer von gr6fieren Welten wieder befal3te Welten" (1: 331).

The recursive structure ofHeinrich uon Ofterdingen is an anticipation of modernity. Brian McHale identifies recursive struc- tures as emblematic of the postmodern novel. "A recursive structure," he writes, 'kesults when you perform the same opera- tion over and over again, each time operat- ingon the product of the previous operation" (112). In termsof chaos theory, his definition describes exactly the mathematical process of iteration which is employed to produce the Mandelbrot set. And models of aperio- dicity, like the Lorenz attractor (see Fig. 4) represent systems such as weather which almost, but never quite, repeat themselves. Gleick writes that Lorenz's "map" of aperiodic systems

always stayed within certain bounds, never running off the page but never repeating itself, either. It traced a strange,

distinctive shape, a kind of double spiral in three dimensions. . . . The shape signaled pure disorder, since no point or pattern of points ever recurred. Yet it also signaled a new kind of order. (30)

Heinrich uon Ofterdingen's recursive struc- ture suggests a spiral in which each circular motion reinforces and "iterates" the next, as dreams recapitulate tales, which echo Heinrich's own life. Each recursion, each mirroring subtly adds substance so that the ontological levels of the novel (tales, dreams, allegories, and action) arenestedwithineach other to form a spiralingwhole. The spiraling structure of Novalis's novel is expressed in several important ways. The first page of the novel describes Heinrich's dream of "die blaue Blume" and of a paradisiacal, har- monious world, full of animals, trees, and rocks which commune with humans. The dream prefigures Heinrich's experiences at the novel's (unwritten) conclusion, in which, according to Tieck's "Bericht," people, animals, plants, rocks, stars, elements, tones, and colors would come together as one family, and would "handeln und sprechen wie Ein Geschlecht," and where "die blaue Blume" turns out to be Heinrich's beloved Mathilde (1: 368). Moreover, Heinrich's dream is precipitated by the Fremdling, whom he meets again at the novel's projected conclusion, another device which reinforces the spiraling of time and "die geheime Verkettung des Ehemaligen und Kiinftigen"

(1: 257-58). This time spiral underlies the entire novel and is reflected in such symbols as the book which Heinrich discovers in the hermit's cave. This remarkable book is not only a book within a book, but it is also Heinrich's life, written in a language which he can't understand, and, like Heinrich uon Ofterdingen itself, is a fragment lacking a conclusion.

Similarly, this spiraling is reinforced by recurrence within the narrative. Novalis repeats not only symbols (such as the color blue, or the notion of a veil) throughout the work, but he also introduces the same themes and characters with each new episode or tale. For instance, the princess in the merchant's tale is a prefigurement of Mathilde, who is later transformed into both the Morgenlanderin and Cyane (1: 368).InTieck's"Bericht,""alles flieRt in eine Allegorie zusammen" (1: 369), in which Heinrich's own mother becomes Ginnistan (Fantasie) from Klingsohr's "Marchen," Klingsohr himself becomes the King of At- lantis from the merchants' tale, and the others similarly assume multifaceted iden- tities from various ontological levels. Thus, Heinrich uon Ofterdingen may be viewed as a prototype of the postmodern novel, described by McHale aspossessing a recur- sive structure in which narrative levels, each with a different ontological value, are nested like Chinese boxes, one inside the other, to the extent that one is often uncer- tain by the novel's conclusion in exactly which ontological level one is (113). The same may be said for Heinrich uon Ofter- dingen, in which ideal and real, dream and Alltagsleben., tale and action were to merge in the end, according to Tieck, so that on- tological levels would be blurred.12

Chaos theory and Novalis's Heinrich uon Ofterdingen share a mutually illumi- natingdynamic paradigm which, in opposi- tion to trends in Enlightenment science, emphasizes the wholeness of natural forms and processes instead of their reduction into useful, static components. There is also a striking prevalence of metaphors of flow in Heinrich uon Ofterdingenl3 which is echoed in chaos theory's emphasis on reality as process. The Mandelbrot set may be under- stood as a dynamic, rather than static, model of reality because it is based on an equation which is infinitely (re)iterated, rather than solved. Scientists themselves at times become mystical in describing the dynamic and turbulent nature of chaos. In a remarkable book entitled Sensitive Chaos, which combines scientific observation with anthroposophism, Theodor Schwenk bemoans the reductionism of Enlighten- ment science. He writes:

Men gradually lost the knowledge and ex- perience of the spiritual nature of water, until at last they came to treat it as merely a substance and a means of transmitting energy, At the beginning of the technical age a few people in their inspired con- sciousness were still able to feel that the elements were filled with spiritual beings. People like Leonardo da Vinci, Goethe, Novalis and Hegel were still able to approach the true nature of water. (9)

As can be seen from his overt reference to Novalis, Schwenk's "Romanticism" is no coincidence. The title of his book, in fact, was inspired by Novalis's aphorism: "Das Fliinige isolirt das Feste, und reciproce. Das F[liiBige] ist wohl nicht Korper zu nennen--es ist das sensible Chaos" (3: 100; Schwenk 9).14

In the end, then, it is possible to conclude that modern chaos theory has much in com- mon with the Romantic paradigm of chaos in Novalis. This accounts for the ways in which Heinriclz won Ofte~~dingen

may be in- terpreted with the aid of modern chaos theory, for the Romantic valuing of chaos and of natural process, and for the flights of language sometimes found in the works of chaos scientists. Sounding more like a Romantic poet than a stereotypical scien- tist, Jacques Cousteau, in his introduction to Schwenk's book, writes:

All around us there arose from the living sea a hymn to the "sensitive chaos.". . . All that life around us was really water, modelled according to its own laws, vi- talised by each fresh venture, striving to rise into consciousness. (8)

This anthropomorphic, analogical language is close to the language of poetry, and it is no surprise that chaos theorists sometimes resort to it to express the dynamic, flowing processes of nature which are at the heart of their research. For, as Novalis remarked: "Unsre Sprache ist entweder-mechanisch-atomistisch-oder dynamisch. Die acht poetische Sprache sol1 aber organisch Lebendig sein" (2: 440).

There is yet one further way in which chaos theory may illuminate Novalis, and

that is in its relevance to the early Romantic agenda, i.e., to the quest for poetic auton- omy. Fabel (Poesie) in Klingsohr's "Mar- chen" asserts: "Ich stehe fiir mich" (1: 301). Novalis's vision of the transformation (tran- substantiation) of reality by autonomous Poesie was shared by Friedrich Schlegel, who wrote: "Sie allein ist unendlich, wie sie allein frei ist, und das als ihr erstes Gesetz anerkennt, dalj die Willkiir des Dichters kein Gesetz iiber sich leide" (183). I suggest that the Mandelbrot set is a visual picture of the Romantic vision of the autonomy of Poesie. As we have seen, Novalis argued that Poesie, weaving together order and dis- order, must reflect a fundamental aware- ness of chaos. But how can Poesie arrive at autonomy when it exists within a world governed by natural laws? The Mandelbrot set, as the visual model of an endlessly iterated equation, manifests fractal images which repeat themselves in no discernible sequence so that the set represents achaotic system which possesses a certain deter- mined, yet unpredictable order. Seeing the full possibilities of the Mandelbrot set would require making literally endless iterations. Like Schlegel's depiction of Poesie, then, the Mandelbrot set is "un- endlich." It is free in the sense that its future behavior cannot be predicted; yet its order is deterministic (the form of the equation never changes). This apparent paradox sug- gests that chaos provides a model which might also shed new light on the perennial debate regarding free will and determinism (Gleick 251). Like Zufall in life and Poesie, the next image in the Mandelbrot set is al- ways a response to the unpredictable "ini- tial condition7-the next iterated number, the next analogy, the next perception. Thus, the Mandelbrot set is an image which il- lustrates Schlegel's formulation of the mis- sion of early Romanticism, for the poetic possibilities are endless, and the whim of the poet is subject to no external law even within a deterministic and finite world.15

WALKER:

Novalis

Notes

l~ee,for example, Mahoney's remark: UNur sollte dabei nicht vergessen werden, dd Novalis nach sehr intensiver BeschaRigung mit Philoso- phie und Naturwissenschaft seine bedeutsamsten poetischen Werke geschrieben hat, sich dieses Vor- gangs wohl bewunt war und ihn als notwendigen Teil der 'Lehrjahre der Bildung'betrachtete" (9).

ZHowever, it is important to note that Novalis's ideas have deep roots in 18th-century thought. Novalis's debt to Kant and Fichte, as well as his own unique synthesis of their ideas, has been amply demonstrated in the critical literature. John Neubauer, for instance, anchors Novalis's "'con- structivist' approach to nature" (131) in Kant and Fichte, but concludes that Novalis's conception of science as "thought experiment" is closer to modern science (especially modern physics) than to the Naturphilosophie of his day (13&37). Also, Gza von Molnir describes Novalis's integration of science, ethics, and poetics as emerging from the Kantian tradition: 'The interrelatedness of ethics and poetics from a Kantian perspective had already been proclaimed in theory and practice by Schiller, and other recognized authors ofthat time, notably Goethe, may also be cited in this respect. Novalis' inclusion of science as essential rather than ancillary to his literary productivity is excep- tional, however, and must be regarded as its most original feature that stands unmatched by his con- temporaries. The singularity of this accomplish- ment is not to be thought of as an aberration or eccentricity but rather as the full envisionment of the entire scope open to modem poetics from the novel perspective granted by the level of conscious- ness Kantianphilosophy had madepossible" (125). Since the grounding of Novalis's ideas in En- lightenment thought has been well established in the critical literature, the intention here is to sharpen the distinction between that practice in Enlightenment science which privileged order and law (as epitomized in Newton) and Novalis's insis- tence that chaotic and disorderly phenomena no longer be marginalized.

3Gleick quotes physicist Norman Packard, who told him in an interview: 'The phenomenon of chaos could have been discovered long, long ago. It wasn't, in part because this huge body of work on the dynamics of regular motion didn't lead in that direction. But if you just look, there it is" (251). However, it should be taken into account that the previous limits of technology (i.e., the nonexistence of computers and even calculators) may have pre- cluded an earlier development of chaos theory because of the complexity of the iterative mathe- matics involved.

4See also:"Believers in chaos . ..feel that they are turning back a trend in science toward reduc- tionism, the analysis of systems in terms of their constituent parts: quarks, chromosomes, or neu- rons. They believe that they are looking for the whole" (Gleick 5).

5"The classical idea is quite certainly that one can break down any object into small, virtually homogeneous parts. Far from this concept being imposed by experience, I will almost dare to say that it corresponds to it rarely. My eye searches in vain for a small area which is 'virtually homogene- ous'on my hand, on the table where I write, on the trees, or on the sun which I perceive through my window" (my translation).

61t is worth remembering that this difficult-to- translate term includes meanings such as "contin- gency," "accident," "coincidence," and "chance." Duden 10 defines Zufall as "etwas, wofiir keine Ursache, kein Zusammenhang, keine Gesetzma- Bigkeit erkennbar ist."

'Though the full complexity of Zufall in Novalis is beyond the scope of this study, it is nev- ertheless a fascinating topic for further consider- ation in relationship to chaos theory. See theSach- register of Novalis's Schriften (5: 815) for a survey of his use of the term.

8Stuart Peterfreund's study of organicism and energy provides a slightly different slant on Fei- genbaum's work, which Peterfreund characterizes as "an attempt to understand the mathematical means of describing energy . . . and the shape of the order that energy may give rise to" (113).

%or has it escaped John Neubauer, who argues: 'The reverse side of this [Romantic] meta- phoric 'scientification' was to conceive the practice of science as a fundamentally symbol- and meta- phor-constructing activity: if poetics and poetry may be seen in terms of metaphors borrowed from science, science itself is a general 'poetic' activity because it perceives and organizes the sense-data by means of metaphors borrowed from elsewhere. Though these metaphors need not be borrowed from poetry, their constitutive role in knowledge makes science an imaginative enterprise. This, I believe, is the common romantic platform and the premise for their attempts to synthesize literature and science" (129).

l0''In its effort to describe the world, science can only proceed by means of a series of images or models which are increasingly 'realistic.' The sim- plest have been continuous and perfectly homoge- neous, like a wire or a cosmos of well-defined and uniform density, or a fluid whose temperature, density, pressure, and speed are equally well- defined and uniform. But it is in other domains where the reality with which one deals reveals itself to be irregular in the extreme, perhaps even to infinity: to such a point that the continuous, perfectly homogeneous model deceives" (my trans- lation).

llsee Theodor Schwenk's Sensitive Chaos for a discussion of these similarities and some striking photographs of natural systems which illustrate the bifurcations characteristic of chaos.

12critics have already explored other aspects of postmodernism in Novalis. See Neubauer for a brief summary of some other contributions to this discussion (130).

13A comprehensive examination of flow in Novalis is outside the parameters of this study, but a few examples from the text of Heinrich uon Of- terdingen may illustrate its prevalence: "der Strom" in Heinrich's dreams (e.g., 1: 278-79); '1m Kriege . . . regt sich das Urgewasser" (1: 285); on love: "Sie istja ein geheimniflvolles Zusammenflie- Ben unsers geheimsten und eigenthiimlichsten Daseyns" (1: 289); and "Daseyns Fluten" (1: 362). See also Peter Kapitza's comprehensive study Die fiiikro~nantische Theorie der Mischung.

14~udgin~from the frequency with which

Goethe is cited in books on chaos theory, a fruithl

study could be made of the roots of chaos theory in

18th-century German literature. See, for example:

Gleick, 163-65, 197; Peitgen and Richter, 3, 21;

Schwenk, 9 and appendix.

151t would be intriguing to examine the impli-

cations of this paradigm in light of Kant's concepts of pure reason, practical reason, and judgment as summarized in his introduction to Kritik der Ur- teilslzraft (242-74). Gza von Molnir has already demonstrated the revolutionary nature of Novalis's poetics (his theory of language) with respect to Kant's distinctions between theoretical and practical reason (12622). An argument might also be made that chaos theory collapses these dis- tinctions, attributing to science (the realm of "pure reason") the "freedom" characteristic of the realm ofpractical reason. Furthermore, chaos may make science more like aesthetics, since new datum (Zufa,Zl)may alter our capacities and thus render science subject to a process similar to aesthetic judgment, which concerns itself with the fit between our affective capacities and the present- ing datum. (I am indebted to Professor Karl Menges for his specific assistance on this point re- garding Kant and for his general encouragement in the writing of this essay.)

Works Cited

Dick, Manfred. Die Entwiclzlung des Gedankens der Poesie in den Fragmenten des Noualis. Bonn: Bouvier, 1967.

Gleick, James. Chaos: Mahing a New Science. New York: Vlking Penguin, 1987. Kant, Irnmanuel. Werlze.Ed. Wdhelm Weischedel. Vol. 5. Wiesbaden: Insel, 1957.

Kapitza, Peter. Die fiuhromantisch Tlzeorie der Mischung: ~ber den Zusa.,n,nenha,ng uon ro- mantischer Dichturagstheorie und teitgenossi- scher Chemie. Munich: Hueber, 1968. Kiipper, Peter. Die Zeit als Erlebnis des Noualis. Cologne: Bohlau, 1959. Mahl, Hans Joachim. Die Idee des goldenen Zeital- ters im Werk des Noualis: Studien zur Wesensbe- stirn~nung der fi.iihro~nantischen Utopie ur~d zu ihren idee~~geschiclatliclzeraVoraussetzu7agen. Heidelberg: Winter, 1965. Mahoney, Dennis F. Die Poetisierungder Natur bei Noualis: Be~oeggriinde, Gestaltung, Folgen. Bonn: Bouvier, 1980. Mandelbrot, Benoit. Les objets fractals: forme, hasard et di~nensiora. Paris: Flarnmarion, 1975. McHale, Brian. Postmodernist Fiction. New York: Methuen, 1987. Molnir, GBza von. "'What Ever Happened to Ethics?"'Literature and Science as Modes ofEx- pression. Ed. Frederick Amrine. Dordrecht: Kluwer, 1989. 113-27. Neubauer, John. ''Nature as Construct." Litmature and Science as Modes of Expression. Ed. Fred- erick Amrine. Dordrecht: Kluwer, 1989.12S-40. Novalis. Schriften. Eds. Paul Kluckhohn and Richard Samuel. 3rd ed. 5 vols. Stuttgart: Kohl- hammer, 1977-1988. Peitgen, Heinz-Otto and Peter H. Richter. The Beauty ofFractals: hnages of Complex Dynwni- cal Systems. Bonn: Springer, 1986. Peterfreund, Stuart. "Organicism and the Birth of Energy." Approaches to O~pzic Fo~,rn:Permutdions in Science and Culture. Ed. Frederick Burwick. Dordrecht: Kluwer, 1987. 113-52. Pfotenhauer, Helmut. Erlauterungen. Heinrich uon Ofterdingen. By Novalis. [Munich:] Gold- mann, 1988. Schlegel, Friedrich. Kritische Friedrich-Schlegel- Ausgabe. Ed. Ernst Behler. Vol. 2. Munich: Schoningh, 1967.

WALKER:

Fig. 3. The Koch Snodake (Gleick 99)

"A rough but vi orous model of a coastline," in Mandelbrot's words. To construct a Ifoch curve, begin with a triangle with sides of length 1.At the middle of each side, add a new triangle one third the slze;

and so on. The length of the boundary is 3 x %.x4/5 x 4/5 ...-infinity.

Yet the area remalns less than the area of a clrcle drawn around the

original triangle. Thus an infinitely long line surrounds a finite area.

Fig. 4. The Lorenz Attractor (Gleick 28-29),

This magical image, resembling an owl's mask or butterfly's wings, became an emblem for the early explorers of chaos. It revealed the fine structure hidden within a dis- orderly stream of data. Tradition- ally, the chan 'ngvalues of any one variable coulrbe displa ed in a so-called time series (top). &show the changing relationshi s amongthree variables require$a different technique. At any instant in time, the three variables fur the location of a point in three-dimensional space; as the system changes, the motion of the point represents the continuously changingvariables.

Because the s stem never exactly repeats itself, t%e trajectory never intersects itself. Instead, ~t loops around and around forever. Motion on the attractor is abstract, but it conveys the flavor of the motion of the real system. For exam le the crossover from one wing oPth~ at-tractor to the other corresponds to a reversal in the direction of spin of the waterwheel or convecting fluid.

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