Prologue: Chasing Traces; or, The Blind Spot of a Discussion
One day in the late 1980s the mathematician Alain Connes and
the neurobiologist Jean-Pierre Changeux sat down together in Paris to hold
“conversations on mind, matter, and mathematics.” Despite being eminent men of
knowledge, they were utterly unable to come to an agreement. The two
adversaries were discussing a fundamental dilemma, the ramifications of which extend
far beyond the discipline of mathematics: Can the world of thinking be
separated from the world of things? And if not, how can we conceive of the
interactions between the two realms? Does thinking—together with mathematical
objects—have a material foundation? Does it leave traces? And are thoughts
dependent on the instruments that make them possible and give them their shape?
Connes argued that there was a mathematical world “independent of us, which
escapes sensory apprehension.” Changeux’s position was the opposite: “Granting
your position for the sake of argument, I’ve tried to see where this world could
be, to see what evidence of it there is
in nature. If you form the hypothesis that this mathematical world exists
outside of us, and if you call yourself a materialist, then it seems to me
you’re obliged to give it a material basis.” The men could not agree.
What is striking about this
conversation, however, is not that the two men would be in the grip of
diametrically opposed ideas. After all, for centuries people have been disputing
such things. No, what was striking was what they left out: Never once was the topic
of mathematical images raised in any detail—and
it was precisely this topic that might have shown them the way to mutually acceptable
answers. In this essay, I will attempt to draw attention to the ways in which
images and objects have either triggered mathematical thinking or served as its
subject matter and helped to shape it.
In what follows, I will touch on
three moments in the “personal-pictures” history of fractals and chaos studies
as modes of visual thinking, which is being presented here for the first time.
The analysis of the private working materials of Benoît Mandelbrot stands at the
beginning of this essay, taking up the greatest part, to be followed by two briefer
discussions of scientists from the related fields of dynamical systems and chaos
theory, Adrien Douady and Otto E. Rössler. Along the way, the surprisingly complicated
relationship between mathematics and its images and representations will be
explored, along with two different meanings of “mathematical islands,” one
figurative and one abstract, each pointing to an antipodal power of visual
representation in the thinking process. Pictures can release a productive,
imaginative power that advances thinking, but attached to them is a seductive and
potentially ungovernable energy as well, one that causes frictions between
vision and thought and even, potentially, egregious errors. These two sides of
pictures are inseparable and equally necessary.
If all three scholars used images
in fundamentally different ways, and if the juxtaposition of their different
methods of deriving mathematical ideas from the world of the visible and the
material will help distinguish three basic different modes of visual thinking, it
is equally true that the thinking of all of them was deeply grounded in visual representations
and that their work would not have been conceivable without images. For all of
them, experiments with early computer visualizations proved decisive, though hand
drawing and other modes of visual and material manipulation were equally
indispensable as modes of thinking and understanding in the realm of
mathematical ideas.
The Discovery of the
Island: Playful Sequences of Similarities
When seeking new
insights, I look, look, look, and play with many pictures. (One picture is
never enough!)
—Benoît Mandelbrot
Viewers fortunate enough to view Benoît Mandelbrot’s first fractal
computer animations, made in the early 1970s, may well experience a jolt of
surprise at the sudden confluence of the worlds of science and science fiction
when the words “The Island of Dr. Mandelbrot” appear on the screen for a few
seconds. Produced by Mandelbrot and programmer Hirsh Lewitan at IBM on a
Stromberg Datagraphix 4020, the animations feature the jagged objects that
Mandelbrot began referring to in 1975 as “fractals”: collections of dots,
lines, surfaces, or solids, “loosely characterized as being violently
convoluted and broken up,” that cannot be comprehended by means of Euclidean
geometry. The films depict such structures as animations generated by chance
algorithms. One sees three-dimensional compositions—dots, lines, the relief
structures produced somewhat later (his so-called “imaginary
continents”)—slowly rotating in front of a black background.
In the line below the intertitle
“The Island of Dr. Mandelbrot” is proof that Mandelbrot himself associated his
fractals with the 1896 fantasy novel The
Island of Dr. Moreau. There, he added in tiny, barely legible letters,
“(with apologies to H. G. Wells)”. With this twist on the title and the mock
“apology,” Mandelbrot deliberately established a connection between “science
and science fiction” and, with gentle, endearing irony, related his own work to
that of a long line of scientists seen as god-like, if weird, creator figures.
There is a hint in these first animated simulations of a classic dream of the
early pioneers of computer graphics, namely that they might, with the aid of
the computer, create a new world, a second reality. By way of the film’s
montage technique, that world was given a quite specific structure, one
expressed in repeated, rapid zooms. The viewer is drawn in closer and
closer—without ever landing— as greater and greater enlargements appear. Zooms
first entered film history as a dramatic effect in Hitchcock’s Vertigo (1958), and ten years later they
were employed by Charles and Ray Eames in the first version of their famous
film Powers of 10 (Rough Sketch, 1968) as a way of staging
a journey between atoms and galaxies. Mandelbrot knew the Eameses, who also
worked in association with IBM, but he characterized their work as “violently
anti-fractal propaganda.” Unlike the designers’ films, Mandelbrot’s did not
emphasize the differences between levels of magnification; instead, as he was
at pains to point out, they created a cosmos out of similarities. The sections
marked with squares in the film revealed in increasing close-up the same frayed
structures as could be seen from farther out. In this way, Mandelbrot
dramatically illustrated one of the most important geometric characteristics of
fractals—their self-similarity.
Perhaps the two key features of
Mandelbrot’s picture making—as well as that of other scientists working in the
field— were his attention to similarities and the works’ appearance in a
series. Mandelbrot routinely worked with sequences of pictures; virtually every
shape appeared again and again, with the very slight differences between them
coming only in the form of minimal adjustments to definition, slight shifts in
picture details, tiny modifications of parameters, et cetera. There were piles
of variant picture shapes that were so similar as to be virtually
indistinguishable. That this is characteristic of pictures from this field of
study raises the question of how picture sequences and single pictures relate
to one another. Can our observation of the material evidence of the working
process tell us anything about how serial pictures facilitate mathematical
understanding? The pictorial work of Benoît Mandelbrot provides the first
answer to this question.
In the Beginning Was
the Image; or, In Praise of a Medium Without Touching It
To see is to believe.
—Benoît Mandelbrot
In 1985, when Columbia University awarded Benoît
Mandelbrot the Barnard Medal for Meritorious Service to Science, it did so with
the following justification: “In the great tradition of natural philosophers
past you looked at the world around you on a broader canvas.” Within
roughly ten years after his first computer animations, Mandelbrot had managed
to create a geometry that brought him renown far beyond the boundaries of
mathematics—a geometry associated with a specific worldview. How did he himself
view the world? Or perhaps better: How did the public in 1985 view Mandelbrot’s
view of the world?
In a press photograph published by
IBM around the time, a smiling Mandelbrot can be seen leaning on one of the
firm’s computers. Behind him, there is a large paper board on which is
written the formula for a general definition of the fractal dimension (D).
(This dimension is an index for the varying degree of roughness of fractal
structures. It quantifies its complexity as a ratio of the change in detail to
the change in scale. The dimension D is always a decimal number and
distinguishes fractals from Euclidean geometry, in which there are only
whole-number dimensions for lines, planes, and solids.) The word “example” also
appears on the board, but the rest of the line is obscured by Mandelbrot’s
right shoulder, and thus the role of the example is taken up by a seemingly
cosmic scenario on the computer monitor on which Mandelbrot rests. Programmed
by his colleague and friend Richard Voss and bearing the title “Planetrise over
Labelgraph Hill (Souvenir from a Space Mission that Never Was),” the image
presents a sphere that looks much like Earth—its colors had been manipulated to
create this effect—looming up out of the darkness of the monitor above a jagged
and shadowed hilly landscape covered with round craters. It is one of the highlights
of the use of fractals for pictorial fantasizing for which Mandelbrot became known
to the public. Shifting the formula literally into the background, it not only presents
the picture the formula produces, it illustrates the prominence of images in
his way of thinking and working—specifically digitally produced images. Indeed,
the decision by the IBM photographer to stage this photo as he did makes
Mandelbrot’s earlier, quite non-public reference to his work as an imaginary
“Island of Dr. Mandelbrot” seem almost visionary. Evidently, the dream of being
able to create a second nature with the aid of fractals, and thereby to have deciphered
mathematically the principle behind the creation of the world, was now Technicolor
reality.
The view of the world that
Mandelbrot presented on his monitor had unmistakable precursors in pre-digital
picture history. Not without reason does it recall scenes from one of the first
American science-fiction classics, Destination
Moon (1950). And even though Mandelbrot gave out that the IBM photo showed
a “space mission that never was,” such flights were indeed taking place quite
frequently then—in movie theaters. Immediately after the publication of
Mandelbrot’s first English-language book on fractals, filmmakers began usurping
the fractal simulation of nature. It was not long before the first planetary
scenes wholly produced by means of his fractals appeared, in Star Trek II: The Wrath of Khan.
Mandelbrot’s fondness for
algorithmically simulated cosmogonies with a science-fiction aesthetic, along
with his frequent use of the provocative phrase “to see is to believe,” greatly
influenced the way his work was perceived. In later statements, he appeared
to glorify visual representation. In a subtle attack on the Old Testament proscription
against graven images, he rephrased the opening line of the Gospel of St. John:
“Wouldn’t ‘In the beginning was the word’ have to be replaced with ‘In the beginning
was the picture?’” Mandelbrot’s unabashed devotion to pictures was not universally
shared; it occasioned heated criticism both within mathematics and outside it.
In their eternal repetition and similarity, images of fractals and chaos were seen
by art historians as breaking with classical mimesis theory, as threatening to
collapse “the difference between picture and reality itself.” It is
significant, however, that these appraisals were primarily based on popularized
images that were not produced in the process of mathematical research. They
were also reactions to the influence the reception of such pictures was having on
postmodern Western discourse: Fractals became a flagship of simulation theory
and a symbol of “new” or even “immaterial” picture mediums that were
categorically difficult to understand and seemed to break with all pictorial
tradition.
Given his close identification with
the computer, it may seem paradoxical that Mandelbrot was wholly incapable of programming
one himself. All his computer graphics were team efforts. Not only did he
gather about him a host of programmers, he also seems to have been conflicted in
his relationship to the device. All his manuscripts and text corrections were either
written by hand or typewritten. He avoided touching a computer himself—an
evasion that was well known in the firm, and that led the IBM photographer to
jokingly comment that he had obviously done so in the photo. In good humor,
Mandelbrot corrected him, claiming that he had leaned against the monitor, to
be sure, but had not “touched it with his
hands.” The anecdote may only be embroidered oral history, but it
touches upon something fundamental: There is a significant discrepancy between
the way scientists employ images and the way those images are seen by the
public. As the present exhibition aims to show, in cases where discussions are
based on popularly known imagery, as was long the case for fractals and chaos
studies, this gap can be immense. Indeed, it provides both material and
motivation for the reevaluation of the role of computers as interfaces between
the world of thinking and the world of things. On the other hand, the
surprising appearance of literary references in Mandelbrot’s newly-discovered
early films indicates a second fundamental and opposing dynamic: It was not
only in the popularization of such images that they came to be laden with meanings that
transcended mathematical or physical issues. Playing with images—with sequences
of similarities—has to be understood as a genuine part of research and thinking
in general.
Mandelbrot’s focus on visual
representation raises questions that can only be dealt with after more detailed
analysis of both his work as a whole and his use of images in particular from
the perspective of image theory and epistemology. His equation of seeing and
believing is doubtless unproven, equally indefensible whether one dismisses it
as evidence of a naive faith in images or takes it as either a rhetorical
flourish or a deliberate, tongue-in-cheek provocation. It still stands as a
polarizing, rhetorically pointed reaction against what he saw as the
devastating effects of certain late-nineteenth-century developments in mathematics.
Mandelbrot was firmly convinced,
and never grew tired of emphasizing, that
the use of computer graphics is now
in the process of altogether changing the role of the eye. The hard theoretical
sciences had banished the eye for a long time, and many observers used to
believe, and even hope, that it would remain banished forever. But computer graphics
is bringing it back as an integral part of the very process of thinking, search,
and discovery.
Where had “the eye” gone, and how had it disappeared?
The Picture as Menace
I see it, but I don’t
believe it.
—Georg Cantor
The progress of mathematics is not always smooth. At the end
of the nineteenth century, for example, the French mathematician Charles
Hermite reported to his colleague Thomas Jean Stieltjes that calculus was
afflicted with a “curse” and that he had turned away in fear and horror “from the
lamentable plague of functions without derivatives.” Mathematics had been
in the midst of a crisis for some time, one brought about by deviations from
classical assumptions about elementary concepts. At the core of the crisis
were functions whose values sprang endlessly back and forth between two
constants and for that reason could not be precisely drawn. Such imprecise
constructions had previously been banished from mathematical discourse, but
even so, they now threatened to lead “to a frontier of the intellect.”
These oscillating, interdimensional
structures could have been a further reason why in the mid-1970s—at the very
moment he was coining the word “fractal”—Mandelbrot chose to make the allusion
to H. G. Wells’s fantasy novel: The fictional Dr. Moreau had tried to create a
new world with man-animal hybrids, and it was for a similar purpose that Dr.
Mandelbrot was using those mathematical hybrids from the nineteenth century. Henri
Poincaré, among others, had referred to these new functions as “monsters” of mathematical
logic and labeled them outrageous attacks on the intelligence of his precursors.
Roughly a hundred years later, Mandelbrot touted these so-called “monsters” as
the first mathematical fractals. He liked to repeat this blasphemy as a way of
pointing to his own achievement in having made out of “monsters” a “geometry of
nature” that, above all, have “the advantage of being beautiful.” By
repeatedly emphasizing that he had incorporated monsters into his new, digital
nature, he was also aligning himself with classical sixteenth-century natural
history, in which nature itself became the realm of “exceptions, the rare, and
the marvelous.”
The crisis in mathematics was also
one of seeing. When the mathematician Georg Cantor questioned the principles
behind the Euclidean idea of dimension, he rightly associated the shock his
discoveries had given him with a disbelief in the reliability of his vision: “I
see it, but I don’t believe it.” It was as an ironic twist on Cantor’s
comment that Mandelbrot chose his own motto: “to see is to believe.” Indeed,
Mandelbrot accused the mathematicians of the turn of the twentieth century of
“blindness” for failing to recognize the visual qualities and the clear import
of their phenomena. At the heart of the crisis was a growing mistrust of
the cognitive ability of the eye and accordingly of pictorial representations. The
more radically even elementary fields of mathematics broke away from the way
humans conceive of space visually, the more difficult it became for the eye to
follow the discipline, and the more inadequate and misleading even schematic
graphics appeared to be. This “plunge of mathematics into the unimaginable” in
response to new “monstrously oscillating” functions has been referred to as
“iconoclasm” by the media historian Bernhard Siegert.
And yet there are a few surviving
visualizations of the first mathematical fractals: simple line drawings, highly
schematic and directly related to structural steps described in texts. They
record the recognition that the human eye could no longer follow geometry’s experimental
thinking since only a few structural steps led it below the visibility
threshold. These drawings were symptomatic of a historic change in mathematics,
one in which the relationship between geometry and the world of experience,
between thinking and seeing, was renegotiated. On the one hand, what made it
possible for the “monsters” to become the subject of official mathematical
discussion was that spatial perceivability (Anschauung)
had ceased to be a normative principle at the time. Given that at least a
few mathematicians recorded their deliberations in graphic form, it is clear that
a certain value was still accorded to the picture and to seeing. But generally
it was clear that in order for constructions that broke so radically with
fundamental assumptions to be integrated into mathematics, the discipline
itself had to be redefined.
The crisis was soon perceived as a threat
that shook mathematics to the core; in 1921 Hermann Weyl described it as a “Grundlagenkrise” (foundational crisis)
that threatened to destroy the “inner solidity and certitude” of analysis.
A way out of the crisis was finally proposed by David Hilbert with his
redefinition of the “foundations of geometry.” As early as 1899, he showed that
the discipline could be axiomatized, an initial but momentous step toward the accommodation
of the uncertainties caused by an explicit rejection of visualization. Hilbert
had concluded that geometrical figures were unworthy of the mathematician’s “superior
intellect” and as a “practical constraint” were contrary to “freedom.” With
Hilbert, mathematical symbols became autarkical. They no longer meant or
referred to anything outside of mathematics. By the end of the 1920s, the
prevailing view was that “objects of present-day mathematics … are no
longer tangible things; … their structures … are of the most extreme abstraction.
… [There is a] tendency toward intellectualization, a disassociation from everything
material.”
In such a climate, the ground was prepared
for the ascendency of a group of French mathematicians who would concentrate
and intensify the iconoclastic tendencies in mathematics. They radicalized Hilbert’s
formalistic approaches and considered pictures to be not only problematic or inadequate
but a menace to mathematical thinking. Beginning in 1935, they grouped together
under the name “Nicolas Bourbaki.” The Greek-sounding name, which was derived
from that of a general in the Franco-Prussian War, served them as a collective pseudonym,
the reasons for which they never fully revealed. They were united in their
championing of a method characterized by its rigorous structuralism and
formalism: For them, structure as a “treasury of abstract forms” was to be
the true subject matter and medium of mathematics. It would be of crystalline
design, “transparent down to the depths.” With the publication in 1939 of
the first volume of their Éléments de
Mathématique, they became one of the most influential authorities in
mathematical research and teaching.
All of their authorized
publications were similar in one crucial respect: There were no illustrations.
According to the Bourbakist Pierre Cartier, their rejection of pictures was
true to “the Euclidean tradition,” which in their view was originally
non-pictorial. In addition, he stated, most of the members were puritanical,
among them a great many Protestants and Jews—“Old Testament people”—and as such
they were ill-disposed to visual representations of religious and mathematical
truths. Their iconoclasm was unwritten but extraordinarily effective. For
example, the iconoclastic imperative is implicit in the suggestion that
mathematical intuition is of importance but in a “manner wholly unrelated to
ordinary sense perceptions.” Because mathematics was to be thought of as a
“poetry of ideas,” the “original visual content” of mathematical figures needed
to be “consciously excluded.” In line with this call for precision, the
discipline was to be entirely transformed into text.
Linked to this radical rejection of
images was a correspondingly forceful admission of their power. This was expressed
most blatantly by a pictorial symbol shaped like a reversed “S” that the
Bourbakists invented for their publications. In the “directions for use” (Mode d’emploi de ce traité) that
preceded their volumes, they explained its meaning and called it a tournant dangereux (dangerous turn).
Whenever their reasoning threatened to become precarious, they presented the reader
with this hair-pin bend: Just by glancing at the page, the reader would know to
prepare for intellectual hardship or logical shoals. The way the bold, squat
typography of the sign visually cuts through the arrangement of the typesetting
demonstrates that the “iconoclastic” mathematicians conceived of their publications’
layout as a pictorial element in its own right. By understanding and using the
image qualities of the typeface, they were able to convey the importance of the
symbol in an intuitive and immediate way.
Digging deeper into the pictorial
history of the sign the Bourbakists chose, an even more striking level of
meaning surfaces. On a formal level, the mathematicians had resorted to the
centuries-old figura serpentinata as
a primal expression of the creative energy in nature and in art, one that can
be identified as a universal “symbol of all drawings” and “the sum of the
visual.” In 1753, the English artist William S. Hogarth defined it in The Analysis of Beauty as an emblem of
“Variety” and accordingly a symbol of all forms of pictorial expression. Some
two hundred years after that, its rich meanings were still being celebrated and
investigated by artists such as Paul Klee, whose graphic work explores the
active dynamics of convoluted lines as a foundation for both artistic practice
and image-theoretical inquiry. The Bourbakists’ stylized serpentine line
was used to flag spots that spelled the same kind of danger for the “realm of
pure thought” that “imprecise” pictures in general did. Thus they banished
what they viewed as the dreaded destructive power of the visual by means of the
visual. The Bourbakist S-curve became a kind of hex sign that called upon the
magical power of the pictorial in a way reminiscent of the ancient stylized
appropriation of snakes as a cultic means of defending against them. One
may assume that they were unaware of the traditional iconic meaning of the
sign, but even so, it functioned as a thorn in the flesh of pure, formalized
doctrine.
In the following decades,
formalistic representation was pitted against geometrical, pictorial
representation in official mathematical discourse. The dogma of
anti-pictorialism left drastic traces in the lives of certain mathematicians,
but it was becoming apparent that pictures could not be completely banned. In
one example, the mathematician and Fields Medal–winner Alexander Grothendieck
first made a name for himself with works in the Bourbakist mode. But just
before his retirement, he experienced a “pictorial turn.” He not only invented
his own “picture theory of mathematics,” in the course of this change he
literally condemned formalism to hell as being inimical to life. In 1971,
after one of his last official lectures at the University of Bielefeld, in
Germany, he left a drawing in the seminar logbook in which a mathematical
notation introduced by the Bourbakists is consumed in raging flames and
surrounded by demons armed with pitchforks. The drawing was embedded in the
following commentary:
Witch’s Kitchen 1971. Riemann-Roch Theorem:
The “dernier cri”: The diagram [displayed] is commutative! To give an approximate
sense to the statement about f : X → Y, I had to abuse the listeners’ patience
for almost two hours. Black on white (in Springer Lecture Notes) it probably
takes about 400, 500 pages. A gripping example of how our thirst for knowledge
and discovery indulges itself more and more in a logical delirium far removed
from life, while life itself is going to Hell in a thousand ways—and is under the
threat of final extermination. High time to change our course!
That change of course seemed to be promised by the invention
of the computer as an image-making machine. In an ironic turn, the picture-rejecting
method based on Hilbert and elevated to the status of dogma by the Bourbakists
had meanwhile become the theoretical point of origin for the computer. The
language of computer programming was derived from a strictly textual
mathematics, yet with the blossoming of that language in computers it became
possible to convert structures no longer accessible to three-dimensional
representation into a new form of visibility. The capacity of the computer to
continuously repeat a calculation and with each iteration add further detail to
the graphic facilitated the generation of this new kind of imagery, which
before had been seen as betraying human intuition. Because fractal structures
are composed of an infinite number of details, the new technology seemed
especially suited to making such highly complex shapes graspable. Technically
speaking, fractals and chaotic dynamics lack closed analytic solutions, which
means one has to actually count through all the states produced by the dynamics.
A human being is at a loss here, particularly because even the calculation of a
single step is a challenge. However, the computer is able to mimic the
generative process of the dynamics in hand and consequently to visualize
patterns inherent to the equation that could not be seen before. In other
words, the dynamics themselves are generatively produced, much like a dynamical
process in nature.
Mandelbrot held the Bourbakists in
part responsible for his having turned his back on Europe and subsequently on
academia. With the help of the computer, proof would be provided that the
rejection of imagery had been a mistake. It was with a heartfelt sense of
triumph that toward the end of his life Mandelbrot was able to declare that “Bourbaki
is over.”
Spotting Islands, Cartographies
of Chance: Analogy as a Principle of Thought
I see things
everywhere.
The eye is not
specialized.
—Benoît Mandelbrot
Mandelbrot once characterized the aim of his scientific
life’s work as the restoration of a “concrete meaning” of sight in which the seeing
eye is accorded the status of a scientific Instrument. The reason it was
such an effective instrument, he believed, was that it functioned in an
interdisciplinary way. To Mandelbrot, this was owing to one of the main
epistemic merits of seeing: “The problem with calculations is that they become very
much specialized in the different fields. On the contrary, the eye is not
specialized. It is a universal tool.” That each academic discipline
practices its own calculation method and relies on specific knowledge acquired
over centuries often renders it barely accessible even to neighboring
disciplines. This could be alleviated, he insisted, through the cultivation of
seeing, a theory he was fond of citing retrospectively to explain his own
interdisciplinary approach and the universal validity of his fractal theory:
They were, he claimed, achievements of the “unspecialized eye.” To him,
formulas and pictures were inseparable: “[E]very formula evoked in my mind a
completely spontaneous shape.” But this did not imply that the eye of a
“layman” was what was required; rather, he was claiming that seeing had no need
of specialization to function as an instrument of understanding.
But this raises the question of how
“unspecialized” an eye should be—or can be. How does Mandelbrot’s testimony relate
to his use of pictures and theoretical thinking, and what clues are provided by
his working materials? At this point, it is necessary to return once again to
the profound clues provided by his early reference to the Island of Doctor
Mandelbrot, and his interest in islands more generally. Pictures of islands and
coastlines were among the most numerous finds in his office in 2011. This was
not a coincidence. They were crucial to his ability to bridge the gap between
mathematical figures and natural forms that had been created even before it
became possible to generate images with a computer. The properties of the edges
of islands had already been featured in an article he published in Science in 1967 entitled “How Long Is
the Coast of Britain?,” which is now considered the founding text in fractal
geometry. There, referring to the “monster curves” of the nineteenth century,
Mandelbrot argues that coastlines are self-similar and infinite in mathematical
terms, constantly revealing jagged new details under enlargement. In the
article, he continues to refer to such structures as “fractional,” a term he
had adopted from the eclectic mathematician Lewis Fry Richardson, whose
findings he discusses in the article. It would not be until the early 1970s
that Mandelbrot would have access to the technical machinery he needed to
generate pictures from these formulas. Looking back, Mandelbrot often described
how the sight of these first computer-generated shapes was first perceived with
a shock of recognition that operated in a similar manner as a déjà vu:
It is impossible to forget the
sleepless nights we spent producing the pictures with programs that we had
diverted from their intended purpose so as to be able to use them at all… .
Although we hadn’t expected it, relief set in: the curve described the coast of
New Zealand and Bounty Island.
[T]he device was very cumbersome, but
when the shapes came out, what a revelation! The pictures were so poor…
.Even so, there was an overwhelming feeling that what we had drawn was right.
Reports such as this document the “Columbus feeling”
frequently referred to in descriptions of early computer use. In the years
of the first experiments with fractal coastlines, producing pictures took all
night, so that Mandelbrot’s “speckled geographies” were often thrown away the
next morning by computer-room operators who thought them to be misprints.
Moreover, for a long time, it was only possible to generate lines, the edges of
which were composed of the letters O, X, M, and W; uniform surfaces were not
possible. Indeed, some of the islands in the catalogue were later amplified by hand
. Mandelbrot did ultimately publish the black, irregular specks in figure 16 in
his first book, Les objets fractals
(1975). He distinguished between them on the basis of their fractal dimension
values 1.1, 1.3, and 1.9. Of the shapes identified with D=1.3, he commented
that they were the ones that displayed the greatest simulative power. Their
similarities to cartography were hard to overlook: The top island was
reminiscent of Greenland; with a quarter turn, the one on the left looked like
Africa; and after a turn of 180 degrees, one could see New Zealand with Bounty
Island in the mass of specks. In that same year, Mandelbrot published similar
chance shapes in another context, noting that he had now discovered Greece, the
mirrored Sea of Okhotsk, the Gulf of Siam, and western Scotland. Mandelbrot
relied on an associative interpretive logic in dealing with such ambiguity,
cheerfully rotating his images in search of familiar shapes.
Mandelbrot’s interpretation of
chance artifacts can be related to a pre-digital history of seeing in the realm
of art. His method was preceded by a long tradition among artists of exploiting
the creative and inspirational potential of chance patterns. Prominent
examples of this practice can be found as early as the Renaissance: In describing
the proper practice of artists, Giorgio Vasari refers to the early stage of composition
when artists would make a “first drawing … in the form of a blotch,” which should
be considered only as “a rough draft of the whole.” He advises that such
blotchy sketches (Italian: schizzi)
be “hastily thrown off” as a means of testing “the spirit of that which occurs
to him.”
A pictorial technique comparable to
Mandelbrot’s associative interpretation of his visualized mathematics can be
seen in eighteenth-century landscape drawings. The English artist Alexander
Cozens (1717–86) first introduced his method of “blot drawing” in 1759. It
allowed one to develop landscapes out of arbitrarily placed and presumably
nonobjective specks of ink blots. The “blot” itself was “no drawing, but a
collection of accidental shapes out of which it is possible to make a drawing.”
Comparing Cozens and Mandelbrot,
one first notes their differences: Mandelbrot’s silhouette-like, large-scale
specks of color come to resemble monolithic, menacing boulders when compared to
Cozens’s dynamic brushstrokes, which reflect the play of light and shadow and
evoke perspective. Whereas Mandelbrot studied his specks from a bird’s-eye
perspective, Cozens gazes into a three-dimensional landscape—even when the
illusion of space is largely absent in this self-contained scene with “receding
hills” and a sparse sky, as the title suggests. And yet there are astonishing
similarities between the two. Cozens’s “blots” were thought of as
“multifunctional” elements: “One and the same ‘blot’ can lead to different sketches”
and consequently serve as the starting point for different imaginarylandscapes.
Landscapes developed in this way were to be formed “out of the spirit of nature
and according to the principles of nature instead of merely replicating a
portion of nature already formed.” Cozens’s “natural” landscapes could not
be identified with any specific locality, and in this respect his associative
picture logic, based as it was on the interpretation of chance blots, was
comparable to Mandelbrot’s practice. While Cozens aimed for a “reassessment of the
relationship between art and nature,” Mandelbrot was suggesting new
relationships between nature and mathematics.
That Mandelbrot’s images only
vaguely resemble the silhouettes of certain countries was part of their
message. They were not intended to emulate any specific geography, but rather
to evoke the idea or basic principle of nature—to be “true to” it. Indeed, dissimilarities
and discrepancies between picture and object were necessary to set him thinking
and to construct the “natural fractal.” This corresponds to a basic quality of
epistemic images in general as analyzed by the historian of science Hans-Jörg Rheinberger,
who has stated that both the “ambiguity of the representational process” and a
certain “vagueness” of the object were required of such images. By means of
his associative method, Mandelbrot was able to access a realm of free thought.
Only through the image and its interpretation were completely new meanings generated. It was these
that produced the connection between geometry and nature, which would not have
been made at all without the image as intermediary.
But it was not computer graphics in
the first place that suggested to Mandelbrot that he look at chance
distributions and nature together. A good ten years before he had access to
such technical facilities at IBM, he had already begun to train his eye by looking
at traditional hand-drawn statistical representations. With regard to figure
18, for example, he noted in 1963 that if one ignored the picture’s source it
was possible to imagine that one was looking at a “new world.” The image in
question, an “experimental illustration” that plotted on three scales the
results of a sequence of coin flips, came from the two-volume textbook An Introduction to Probability Theory and Its
Applications by the mathematician William Feller. Mandelbrot borrowed Feller’s
illustration without a single change, though he proposed a geomorphological description
of it that added an additional level of meaning: “First of all, forgetting the origin
of that figure, let us imagine that it is a geographical cross section of a new
part of the world, in which all the regions below the bold horizontal lines are
under water.” By using another mathematician’s picture for his own
purposes, Mandelbrot was able to establish a strong and characteristic connection
between chance and nature. The chart came to assume an iconic status for him,
and he reproduced it in all three of his books on fractal geometry, as well as in
a number of articles.
That Mandelbrot’s way of seeing
served to update visual traditions can be discerned in the development of statistical
graphics. For example, the now legendary diagrams of the economist William
Playfair, considered one of the inventors of the field, were designed to be
easily scanned, a quality characteristic of landscape depiction as well.
His Wheat/Wages Diagram from 1822, for instance, relates the price of grain to
the weekly wages of workers between 1565 and 1821. That Playfair wanted the
viewer to register the information with ease is indicated by the last three words
of the phrase he confidently placed in the middle of the image: “Chart Showing at
One View.” The synoptic qualities of the visualization were crucial to him, as
can be seen in his choice of a background for the changes in wages: the trope
of a silhouetted towered city, familiar from early modern cartography, yet
colored in a kind of gray sfumato to reinforce the distance. To obtain the
impression of a landscape, he combined graph and column diagram, two modes of
representation that are very different. Playfair unambiguously if tacitly
answered the importance of the “specialization of the eye”: In learning to read
new picture forms, it is necessary that one be presented with well-established,
familiar visual impressions. Instant legibility—showing at “one view”—all but
requires the use of other genres’ pictorial logic. It is in them that
preexisting visual habits are exercised and a picture’s subject can be accepted
at first glance as logical because familiar. A similar representational mode
was employed in the development of geology as a separate scientific discipline
at around this time. Indeed, a glance at the chart may suggest geologists’
characteristic transformations of mountainous regions into panoramic
crosssections— just one among many examples in history of a mode of
visualization being handed along from one science to the other.
Playfair’s strategy shows that the
eye can accommodate itself to a new picture form if it comes into contact with
familiar ways of perceiving reality, as in an image that has been designed in a
particular way or, as Mandelbrot demonstrated in his reading of Feller’s
diagram, an existing image that has been cleverly re-interpreted. Mandelbrot’s repurposing
of Feller’s diagram allowed him to visualize chance in the jaggedness of the
earth’s surface; in turn, this allowed him to treat the features of a certain
part of nature—the “sculpture of mountains”—as something that could be analyzed
using statistics.
Both Mandelbrot’s first impressions
and the later configuration of his images combined meanings and visual
traditions from different contexts. These not only influenced the dissemination
and popularization of Mandelbrot’s work, they were part of his exploratory
thinking process, shaping thereby his conceptions themselves.
As for the
use of images in the construction of a fractal “geometry of nature,” Mandelbrot’s
“unspecialized eye” sought to figuratively reinterpret representations of abstract
knowledge. Schemata that could no longer be deciphered on the experiential level
were approximated to visual habits; thus they functioned as an experimental field
for new constructions. Indeed, such readings could take on a significance of their
own that affected Mandelbrot’s interpretation of their content.
Mathematical structures that were unrecognizable in visual terms, and
accordingly appeared to exclude the eye as an instrument of understanding, were
objectified, in analogy to geomorphological features, as visually identifiable
spaces.
The way
Mandelbrot saw and interpreted his images tended to break down traditional distinctions
between the use of pictures in art and in science whereby images in science are
supposed to be unambiguous. Instead, as Mandelbrot helped to demonstrate, pictures
in the sciences have to be seen as fundamentally obstinate, intractable instruments
whose vagueness can be productive in research. Scholars such as Mandelbrot have
utilized these qualities of the visual in a playful, associative manner, one
that has led them to new insights.
Many of the first artists to
experiment with the computer in the 1960s sought to use scientific methods,
“with their claim to objectivity, transparency, and verifiability,” to
reconstruct “an aesthetic on a scientific basis.” Mandelbrot took the
opposite approach; he made use of artistic techniques that exploited the
subjectivity of visual impressions to create natural forms out of images of
chance. In this way, a kind of “role reversal” occurred between motives conventionally
attributed to the realms of art and science respectively.
At the same moment, this apparent breakdown
between traditional distinctions also led to new confusions. One has to wonder about
the degree to which this perceived “role reversal” influenced the scientists’ self-image.
Here it is worth noting a seemingly insignificant detail in some of the pictures
in the exhibition: Mandelbrot’s signature. On my first visit to his office in
2008, he insisted on signing all the pictures, individually, and with a
fountain pen, before he allowed me to scan them for my research— seventy in
all. This procedure offered clear evidence of a pronounced sense of authorship—
even for those pictures that had been misprinted, torn, or stained.
Signing a picture is a practice
mainly associated with fine art, a way of claiming responsibility for one’s oeuvre. Mandelbrot candidly admitted
that he played with this ambivalence: “For Who’s Who I put down MATHEMATICIAN
AND SCIENTIST or even MATHEMATICIAN, SCIENTIST, SCHOLAR, AND ARTIST, which
upsets editors, who prefer a single word.” How did he come to think that?
Was this mere hubris or evidence of a misunderstanding?
There would be no point in suddenly
declaring Mandelbrot’s signed pictures to be “artworks.” The question itself is
unproductive, as it tends to lump together the varied pictorial products of art
and science and seduces one into “blurred simplifications.” More to the
point is whether a separation between mathematics and art is justified at all
with regard to pictures and their possibilities— particularly their epistemic
possibilities.
Mandelbrot’s islands provide a
perfect example of how his creative method left its mark on the world of
mathematics. Having spent years testing the effects of various dimensional
values in order to determine how he might achieve the greatest simulation of
“nature,” Mandelbrot arrived at an important mathematical thought solely on the
basis of a visual impression. Looking at some of his pictures, preparing the
manuscript for what was to become Fractal
Geometry, he once again had the sense that he was looking at landforms and
then all of a sudden, “I screamed: Stop it, stop it! … I see an island, I
see a continent. It’s amazing—this is Spain, this is Great Britain … and
then Scandinavia!” Since he had previously used the fractal dimension of
1.3 for such distinct impressions of nature, he now conjectured that these
pictures must also have the value 4/3 (rounded to 1.3). That Mandelbrot’s
intuition was correct was ultimately proved in 2000 by the mathematicians
Gregory F. Lawler, Oded Schramm, and Wendelin Werner. This gap between
hypothesis and proof was not untypical of Mandelbrot, whose creative,
associative way of looking led him to conjectures rather than to rigorous mathematical
statements. Lawler himself compared Mandelbrot’s approach to that of the
archaeologist who digs and makes exciting discoveries but leaves it to others to
provide the explanation or the context—or in Lawler’s words: “If he had not
produced the questions, other people would have not found the answer.”
The creation of visual
analogies—the pursuit of analogical seeing—was the engine that drove
Mandelbrot’s research. Indeed, he came up with the most appropriate symbol for
this method himself. In 1974, having set about creating his own designs for the
cover of Les objets fractals, he
began to cut out illustrations from the book and paste them as a collage on its
front cover. He produced three separate versions of the design, which was not
in fact used, based on his model of the distribution of galaxies, his
speck-like “islands,” and his “imaginary continents.” True to Mandelbrot’s
credo—“before beginning to understand what fractals are, one should know what
they look like”—an overview of the phenomena was to precede even the opening of
the book. On the back cover, a collection of straight lines—a basic element
of Euclidean geometry—was to appear. Mandelbrot identifies this collection of
lines with Parisian streets—but if it is an imaginary city map, it looks as
if it had been struck by an earthquake. The viewer feels as if he or she is
looking through the lines, which are of various widths, into a box of great
depth, while the direction of the streets has been whirled into a confusion that
extends across the entire sheet, an effect generated by the application of
calculated pseudo-chance. Thus, a stochastically transformed “culture” and a
programmed “nature” are juxtaposed and interrelated. In the jacket text,
Mandelbrot states: “[I]t will be suggested that there are far-reaching analogies
between all these objects that are intuitively comprehensible. Despite their novelty,
this must be seen as fundamental.” Mandelbrot himself provided the key word
there: “analogy.” This was a concept fundamental to his thinking. The
juxtaposition of images on his book-cover designs recontextualized his picture
shapes and gave them new meaning. More generally, the tableau as a
configuration of shapes in a common picture space serves to symbolize his
entire approach. This gives a first answer to the initial question of how
mathematical thinking can be organized and make sense given a surfeit of
similar pictorial material in a series. The single picture showing (or rather embodying) the exemplary individual case
becomes more general—demonstrating a way of seeing that creates analogies and
reveals associative potential as part of a multitude—while still keeping its
validity as a unique image. In short, a logic of seeing takes hold that can be
described as the “logic of the tableau.” But it should also be noted that
with his theory that computer graphics alone had led to a “return of the eye”
in mathematics, Mandelbrot underplayed the role of sense impressions in the acquisition
of knowledge and in so doing helped to skew people’s assessment of his own
contribution. On the contrary, it was his imagination, initially developed
independently from the computer, that had inspired his research and served as
an antidote to the dominance of calculation.
The Emptiness Between
Islands: Media and Visibility; or, The Computer and Its Blind Spot
I saw it but I didn’t
know that I was seeing it.
—Benoît Mandelbrot
How visible are islands? Does an island stop where the water
begins? And how can one discover what lies between islands? Such questions may
seem strange at first, yet they lead to the crux of a riddle that was still
unsolved in 1980—and to a second way that islands were important to Mandelbrot’s
pictorial thinking. From imagination and the figurative meaning of the islands
as geomorphological structures, the focus is thus shifted to their abstract
counterparts: “islands” and “continents” as metaphorical terms that Mandelbrot
used for decisive pictorial details on his computer-generated prints when he
was about to make one of his most well-known discoveries, the Mandelbrot set.
These components, and especially the mysterious spaces between them, raised
serious issues about the mutual dependency of the world of mathematics and the
image technologies that generated them. It was the riddle of those islands and the
question of whether they were really peninsulas that was plaguing Mandelbrot and
that revealed the twofold, antagonistic power of the visual for thought: the
potential to seduce the mind through mere appearance. As the pictorial history
of this famous symbol of chaos theory demonstrates, vision and thought had by
no means always been perfectly in harmony. Rather, seduction can be determined
as a second force that coexists with imagination. Imagination and seduction are
two sides of the same coin in visual thinking.
The majority of the pictures from Mandelbrot’s
office on display in the exhibition are related to experiments he carried out
in 1979–80, shortly before his discovery of the set. Yet if these images are
dominated by mere blankness, patterned by a covering of small dots, or
structured by fuzzy shadows or indefinite outlines, the Mandelbrot set became
generally known in a wholly different form, dazzling and brightly colored, as in
the computer graphic that appeared on the cover of Scientific American in August 1985, which looked like a dynamic fireball
racing down a black slope. Above all in the margins, the Mandelbrot set is
distinguished by impressive ornamentality and self-similarity. This is all the
more striking given that the formula it is based on seems simple even to
non-mathematicians. This picture is typical of the popular composition and
distribution of the set, which in the mid-1980s became one of the first
scientific “icons” in the digitalization era.
Despite its shaded, vivid, and
relief-like appearance, the Mandelbrot set is at heart nothing but a series of
numbers that can be graphically represented within the complex number field.
Because complex numbers consist of two components (real and imaginary), the
picture space of a digital graphic that sits in a coordinate system with its x
and y axis allows us to produce a correlation between the invisible equation and
the visible surface: Each pixel on the screen corresponds to one precisely
defined complex number. That said, the discrete, grid-like structure of
digital-visualization technology has also generated its own zones of insecurity
for this correlation: While a non-linear dynamic structure generally takes up an
infinite and continuous space, if it is to be represented digitally, on a
computer, it has to be reduced and approximated. As a consequence, pictorial
details that seem unrelated or “far apart” on an image may well be connected or
“very close” as far as their provable mathematical properties are concerned. In
short, visibility and existence in a mathematical sense—what can be seen and
what can be rigorously proven—are not the same.
The gap between appearance and “reality”
became a serious challenge for Mandelbrot’s investigation of his “mathematical islands.”
On the aforementioned 1985 cover, for example, there appears a black bulbous
shape that Mandelbrot would have called a “continent” (i.e., the main, central
part of his set); on the upper level, in a black squiggly blaze, one can make
out a black shape of similar appearance, only smaller: an “island” (i.e., a self-similar
replica). In this image, there are no empty spaces apparent between islands and
the continent. Both are connected visually through various color nuances and
shades. But this was by no means the case during the discovery process. At
first, continents and islands were separated by empty spots. What was the
significance of the transition from empty intervals to color connections? To
explain the mystery of the islands and their problematic visibility—and hence
their image-theoretical relevance—it is necessary to take a closer look at the process
by which they were discovered.
The discovery of the Mandelbrot set
was based on work done by the French mathematicians Gaston Maurice Julia and
Pierre Fatou at the beginning of the twentieth century on the dynamics of
polynomials in the field of complex numbers. Their papers were held to be
extremely difficult masterpieces and as such were long considered all but
impossible to understand, even by mathematicians. Mandelbrot studied under
Julia at the École Polytechnique and knew his work, but until the late 1970s,
he had never concerned himself closely with it. It was only when he realized
that a computer might be used to produce pictures from Julia’s analyses that his
interest was awakened. At a colloquium in Princeton in 1977, Mandelbrot was
made aware of computer graphics by the mathematician John Hubbard, with whom he
made contact a year later. Hubbard was one of the very first people to
produce graphic depictions of cubic polynomials. Though Mandelbrot spoke of
Hubbard’s pictures as being “of rather poor quality,” they must have made
a strong impression on him, as he soon invited Hubbard to IBM to learn about
his ability to generate computer visualizations from Julia’s formulas. This was
not long before he himself, armed with some of the most powerful computers of
the time, began producing pictures of so-called Julia sets, a practice that
would ultimately lead to the definition of the Mandelbrot set.
It is possible to distinguish two
phases in Mandelbrot’s research. In the first, at IBM in 1979, he devoted
himself exclusively to the graphic depiction of complex non-quadratic
equations. In the second, as a visiting professor at Harvard during the winter
semester of 1979–80, he focused under less than ideal technical conditions on
quadratic equations and their illustration. The two phases can be characterized
in terms of sense perception, the first being a phase of seeing and the second being a phase of discovery: “I did see those elements [i.e., what were
later called the small “islands” as part of the Mandelbrot set] in 1979 but
could not organize and describe them. Therefore, they remained undiscovered.”
That his seeing did not take place at the same time as his discovery
contradicts the all-too-simple assumption that the Mandelbrot set simply
suggested itself spontaneously once calculating capacities reached a certain
level. Such a view is no more accurate than the claim that the set’s discovery
would have been possible before the invention of the computer. The history of the
Mandelbrot set can neither be disassociated from the requisite technology nor attributed
to it alone. The main difficulties that stood in the way of its discovery at
the beginning resulted from Mandelbrot’s having to deal with pictures new to
this field, pictures that needed to be observed. With the computer, observation
became a new epistemic category of mathematics.
During the phase of merely seeing
and of playful experimenting with the new shapes, Mandelbrot, together with his
programmer at the time, Mark R. Laff, produced an extensive series of
black-and-white graphics. During my survey of Mandelbrot’s office, I came
across some 150 images from this period which he characterized first as “state-of-the-art,”
then as “used furniture,” and finally as “valuable antiques.” They document
most impressively how comprehensive his experiments were and how difficult it
was to make anything visible at all. Mandelbrot did not publish these pictures at
the time because it was simply not possible to make sense of them or to “clean up
their theory.” At the beginning, it was by no means certain that any of
his experiments would create a clearly recognizable pattern at all. An
equation’s complexity was often reflected in an apparitional blurring of its
contours—or even, as happened frequently, in the fact that the picture simply remained
white, and nothing at all could be seen. Some of the images are immediately reminiscent
of glimpses into distant galaxies. The appearance and disappearance of the patterns
seem to be synthesized in a single picture space.
Whereas the choice of coloring is
arbitrary and depends on the technology used, the assignment of colors—or the
decision to just use black and white—has to follow neatly defined rules that
have to be fixed before the image is generated. These rules identify the range
of values—the threshold— for which a specific color will be assigned. To
generate a graphic, the computer takes each of the numbers in the plane (i.e., pixels),
inserts its value into the formula, and iterates it for a previously defined
number of times to find out what specific color has to be assigned. In chaotic
dynamics, it is a characteristic of the equations to react strongly even to the
tiniest changes, a quality known as “sensitive dependence on initial conditions.”
Consequently, small changes in the parameters of an equation can lead to completely
different behaviors. The technological limits of the resolution entail two major
problems. First, two patterns that are theoretically distinguishable may blur in
the computer graphics and thus appear to be indistinguishable. Secondly, error in
rounding-off induce artificial changes in state, which in turn influence
behavior in the long run because of the equation’s dependence on initial
conditions. Previously distinct shapes and patterns may, therefore, start to
fade and blur. The technology and its limits of resolution determine the degree
of fuzziness.
The difficulty of achieving
precision in the representation of equations explains why, in many of the
images—especially the huge series of 150—only a few outlines are clearly recognizable,
with the majority disintegrating into irregular cloudlike spots: “Throughout,” wrote
Mandelbrot, “I felt like someone trying to photograph the Cheshire cat in Alice in Wonderland at the very moment
it is about to disappear.” The pictures that he compared to the transitory
appearance of the famous mythical feline from Lewis Carroll’s novel reveal the
fragile condition of the number field known among mathematicians since 1918 as
“Lattès
chaos”. The closer the parameter value λ approached the number 1, the
larger the gray areas in the picture became. Like the photographer who
controls his shot by his choice of lens and his use of lighting, Mandelbrot
identified more or less photogenic and picture-worthy motifs in the complex
number field. Instabilities between appearance and non-appearance are
indicative of the fundamental characteristic of non-linear equations. Because
of their great sensitivity to changes in parameters, they lend themselves to
the illustration of the transition between order and chaos in dynamical
systems. What was to be considered order and what chaos depended on the visual
arrangement: For example, gray areas with interference, from which a pattern was
wrested out of multiple variations of the most diverse parameters, belonged to
the classic iconography of chaos. Mandelbrot acknowledged in retrospect that in
places in the series it might already have been possible to recognize the
familiar bulbous shape of the Mandelbrot set. He subsequently glorified these
as “advance-shadows of the Mandelbrot-set-to-be,” which he had not yet identified
as a shape of its own.
The blurring and the empty spaces in
these pictures, however, point to the complex relationship between formula and
image. Visualizations can always only be approximate values of non-linear sets.
Since the calculations inevitably have to be performed with a finite number of
iterations, the resulting images always bear traces of a compromise. The
encounter between formally infinite mathematics and the finite abilities of the
computer owing to its limited processing power necessarily produced indefinite
and hence blurry areas. As a result, Mandelbrot focused on investigating appearances
and disappearances and searching for the right picture moment for recognizing a
pattern, as well as on the different possible meanings of the empty spaces on
the sheet of paper. It was not always possible to determine unambiguously how
the visualization and the mathematics related to each other. Owing to the
technique and its limitations, there were built-in contradictions and areas of
friction between theory and image. It was because of those interstices and
empty spaces that the island issue would soon become acute—and tempt Mandelbrot
into hastily drawing mathematically unproven conclusions.
The problems of visualization became
especially critical when Mandelbrot moved to Harvard. There he had neither an office
of his own nor his own machines to work with. He was dependent on the older
computers in the basement of the Science Center that were available to
everyone, shimmering green monitors with cathode-ray tubes and dated printers.
The brief working time available and comparatively poor equipment forced him to
reduce the complexity of his research subject. Turning away from cubic polynomials,
he decided, together with his assistant Peter Moldave, to experiment with quadratic
polynomials with a single variable parameter. What he expected was a simple, unambiguous
result. Yet what appeared was unlike anything he had experienced before. The
first pictures “looked awful: filled with apparent specks of dust that the
Versatec printer produced in abundance… . I was motivated to sniff out the
ramifications of those specks of dirt”. According to all reports, the
monitor at Harvard was subject to constant malfunctions and the decrepit
printer made it uncertain what one was really seeing. Were the specks in fact dirt
or “dust”? Were they smears and blots of the sort that the machine produced in abundance
or visualizations of mathematics? For the moment, the boundary between image and
accident was blurred. Suddenly, something surfaced that could not be understood
and that raised new kinds of questions: Were the specks part of the picture or
not?
In order to
distinguish between picture and phantom images, Mandelbrot decided to examine
the specks more closely, employing in effect the computer as a microscope. It
was only his perception of symmetries in the “dirt” that elevated the “reality
probability” that the specks belonged to mathematics: “Since bona fide dirt is
not symmetric, both kinds of specks demanded to be blown up for close inspection.
… The symmetric points had more chances of being real.” From a series
of enlarged pictures, it was determined that some of the specks were in fact
dirt from the machinery, but others proved to be self-similar copies of the
basic shape. The presumed glitch was thus transformed into a decisive
discovery.
Mandelbrot dubbed the self-similar “messy”
speck structure surrounding the main body of the Mandelbrot set as “islands”—“offshore
islands”—though sometimes he also referred to them jokingly as “pearls in the
pigsty” because they had been discovered in the dirt. For years he tried
to ascertain their precise structure. The crucial question was whether or not
they were linked to the main structure of the set. The computer images gave
them the appearance of free-floating elements. Yet how did the picture relate
to the theory? Could their mathematical nature be related to the way they
appeared? Were the visible parts of the picture also existent in a mathematical sense? Could he believe what he saw?
Because the small specks were
deceptively similar to interference or printer smudges, they were repeatedly
retouched by conscientious computer-lab and print-shop employees. For that reason,
in the years 1980–81, Mandelbrot got in the habit of posting warnings on his
output equipment: “Do not clean off
the dust specks, they are real.”
According to his own reports, the fact that his early fractal pictures were in
constant danger of being erased had infuriated him on many occasions. Likewise,
his first “chance geographies” had repeatedly landed in his co-workers’
wastebaskets as presumed discards or misprints. With regard to the “reality
status” of his visualized mathematics, a striking reversal can be seen in his
attributions: He now called his simulations of natural forms “imaginary,”
while the visualizations of his complex equations were classified as “real.”
Mandelbrot was fond of insisting that he had created his new geometry “out of
the dirt”—clearly a further jab at the formalistic Bourbakists, to whom “dirt”
was the avowed enemy of mathematical, “crystalline” logic.
Mandelbrot’s unconventional and
creative reliance on his “unspecialized eye” would never have been possible
without pictures. Here, however, it can be seen that his trust in it had its
drawbacks: In his first publication of the Mandelbrot set, in 1980, Mandelbrot
confused the mere “appearance” of the small islands with their mathematical existence.
In his manuscript, he noted: “A striking fact, which I think is new, becomes
apparent here: FIGURE 1 is made of several disconnected portions.” The specks
of islands seemingly detached from the main body of the set were declared, in a
fateful confusion of image and proof, as a “fact,” although this assertion was
clearly unproven, as it was not supported by any mathematical theorem.
Moreover,
observant viewers of the picture published in the “Annals of the New York
Academy of Sciences” could not even make out the specks discussed in the text, for
once again they had been deleted as supposedly unwanted printing flaws by the editor:
“Horrors! It is now free of specks! … Clearly, gremlins in the printing business
had … repeated that evil deed.” In offprints of the article that he
sent to colleagues, Mandelbrot reinserted the deleted picture details by hand.
Since he did not see any connecting lines and had unlimited faith in his
computer, he did not believe they existed, a faith he reinforced in the
drawing. Suddenly, Mandelbrot’s “seeing is believing” had changed to “drawing
is believing”: to trust in the digitally produced shape, which could be—and had
to be—authenticated in drawing to exist. By means of his own successive acts of
reinterpretation, the same pictorial details mutated from a hint to a proof and
finally ended up as the main result: A material trace became mathematical evidence.
The
mathematicians Adrien Douady and John Hubbard provided the counter-evidence in
1982. In an article in Comptes Rendus,
they determined that fine lines did connect the specks with the main body but that
they could not be made visible at that point in history: “These islands are in
fact attached by threads that evade the computer.” Douady compared the
complex problem of making these structures visible with trying to see tiny
mountain rivulets from an airplane flying overhead. When Mandelbrot was
making his experiments, it was technically impossible to draw the connecting
lines, or filaments, with the existing graphic programs. There was a built-in
impenetrability between the two “planes” of the digital image, its visible surface,
and the indiscernible columns of numbers that generated the picture (i.e., the operational
undersurface that processed the formula). Accordingly, an unbridgeable divide
was created between the “islands” and the “continent.” A link between the two
planes was achieved only a short time later through depictions in shades of
gray or colored graphics. Color was able to fill the previously blurry or empty
zones at least visually and to distinguish a variety of different behaviors in
the dynamical system. In consequence, Mandelbrot felt compelled to expand his
earlier slogan, “To see is to believe,” into “To see in color may lead to an even
higher intensity of belief.”
If one were to write a history of
the sciences as a history of pictures, one that told the story of the
construction of the visible out of the invisible, the issue of limitations on
what can be visualized—i.e., visibility thresholds—would necessarily be a
primary topic. At what point does something come into view, when does it
disappear again, and when does one decide that it is “visible enough” to be
perceived as an actual, provable phenomenon—to be judged as a mathematical
“fact”? This touches on basic questions concerning the epistemic function of
technical images and the elementary relationship between picture-producing methods,
pictures, and knowledge. New picture-making methods can open up new realms of
visibility and understanding to the same degree that they can also once again limit
our seeing.
In Mandelbrot’s case, the realm of
visibility also defined the horizon of understanding; or, as the science
historian Thomas Schlich has written regarding the problematics of microscope
pictures, “The limits of understanding coincided with the limits of the
visualization technique.” Since Mandelbrot placed unlimited trust in his
machines and had not seen the connecting lines—could not see them—he had failed
to credit their existence and had confirmed as much again in his drawing after
the unwanted retouching. His own equation of seeing and believing had misled him
into claiming he had discovered a “striking fact” from the mere appearance of the
pictures.
Nevertheless, the story of the
Mandelbrot set is not one of confusing pictures with reality or of subjective
naiveté. On the contrary, it raises the question of how one arrives at
understanding through computer graphics in a paradigmatic way. It clearly shows
how difficult it can be to maintain a distance between what the eye can see and
what can be proved analytically. As the story of a struggle for agreement
between observation and theory, it illustrates equally clearly how necessary it
is for the eye to be schooled on pictures and how risky thinking inspired by
pictures can be. In short, it says something about the fundamental ambiguity of
visual perception.
When one considers all of the
subsequent events, one comes to recognize that the story of the Mandelbrot set
also includes an element of tragedy. Looking back on these events in an article
published six years later, Mandelbrot not only regretted the false assumption
made in 1980 and the “awkward surrogate” of a conclusion that resulted
from it, he also searched for ways to exonerate himself after the fact. One notes
this specifically from his retrospective narrative of the story and from the
caption beneath his subsequent publication of Julia sets in 2004. There he
insists that an analysis of the theory had taught him as early as 1980 that the
images of Julia sets were deceptive, and then he adds: “These graphs were
important to my thinking because they sufficed to show that the broken-up early
M set pictures were compatible with connectedness.” It would be a
simplification to call this surprising statement a mere contradiction. First
and foremost, it emphasizes the intellectual struggles that computer-generated mathematical
images were able to cause and the ambivalence that they triggered. Second, it
demonstrates an attempt to prove after the fact that his thinking was actually
correct despite his previous erroneous assertions. Given the importance Mandelbrot
accorded to the virtues of pictures in all of his work, his visually triggered misinterpretation
in the first publication of the Mandelbrot set in the “Annals of the New York
Academy of Sciences” in 1980 must have sat like a thorn in his flesh.
Dr. Mandelbrot’s islands and the
issue of their visualization played a key role in his work as a mathematician.
On the one hand, they illustrate the virtues of the “unspecialized” eye and
thus of the interdisciplinary effectiveness of visual perception. On the other
hand, they highlight the seductiveness that adheres to computer-generated visualizations
of formulas. In this way, they get to the heart of the question of how pictures
are able to convey knowledge and can be seen as symbolic of the two opposing,
centrifugal forces involved in pictorial thinking.
The formula “Seeing is believing”
fails to give due credit to the particular role that the visual plays in the
process of scientific cognition. Pictures have to raise doubts and stir the
imagination if they are to generate understanding. In that way they can serve
as the raw material for thinking or as thought thresholds for the mind.
The Linear Makes the
Non-linear Visible: Drawing Is Seeing
Drawing is a
discipline of vision.
—Edward Hill
If one wants to pursue the investigation of the
interdependency of the visible sphere and the world of mathematical ideas, it is
especially fruitful to continue to look for interactions between early computer-generated
pictures and hand drawings. As can be seen in Mandelbrot’s work, one of the
capacities of drawing is the authentication of computer-generated results. But
the potentiality of drawing goes far beyond this function. To identify other
epistemic roles of drawing, it is now time to leave the work of Mandelbrot and
have a closer look at the practices of some of his contemporaries as a
comparison, and first of all, at the work of Adrien Douady.
Not all mathematicians who worked
with computers shared Mandelbrot’s conviction that computer graphics had
“brought back the eye” into mathematics. Douady, a French Bourbakist, an expert
in the field of complex dynamics, and the man who first proposed naming the
Mandelbrot set after Benoît, was of the opinion that pictures and the eye had never
wholly disappeared from mathematics insofar as geometry-oriented mathematicians
had been making drawings all along. He reported that even when the dogma of iconoclasm
was in ascendance, the visual had found secret yet effective ways into the discipline.
Mathematicians had been known to climb rooftops in Paris to draw gigantic diagrams
with crayons on walls and to make sub rosa sketches on pieces of paper under
tables during lectures. He stressed that “the eye” had only disappeared
from the sphere of publications (“expression”) but never from the sphere of
thinking and the creation of ideas (“conception”). As an experimental tool, the
computer had revealed new structures, to be sure, figures that one would never
have been able to see otherwise—like all the visualizations of iterated
polynomials that Mandelbrot had produced in 1979—but that alone had by no means
facilitated a mathematical seeing: “Sur cette donnée brute, vous n’y voyez rien.”
(On this donnée brute, you don’t see
anything.) The French term “donnée
brute,” which Douady uses for digital pictures both seen on the screen and
as printouts, can be translated as “raw data”; it refers to pictorial material
that has yet to be processed analytically. A second meaning of the term, the
“raw given,” makes it clear that here the mathematician is operating from the
start not only with data but with something visually concrete.
In their basic building blocks—and because
of the low resolution of computer graphics in the early 1980s—representations of
Julia sets consisted of exploded black rectangles that only outlined raw shapes;
they had yet to be made accessible for seeing, at least in the sense of
mathematical comprehension. Computer-graphic representations of formulas
contained a surfeit of information, which Douady translated in an epistemological
three-step process from a digital to an analog medium in order to interpret
them. One of Douady’s worksheets shows how he first placed the “raw,”
“unpolished” rectangles derived from a formula on a sheet for preliminary consideration
(on the left), then drew connections between some of those tiny rectangles (in
the middle), ignoring others or accenting them with small circles (interprétation). In a third step, seen
at the far right of the picture, he reconstructed and abstracted the
information it obtained in a branching line. That line, employed as a tool in
the analysis of the dynamics of Julia sets, is reminiscent of a curved branch,
sharply bent at the ends (arbre plongé),
that would later come to be known as a Hubbard tree.
Douady’s drawing was an independent
tool for understanding that played both a mediating and a
constructive-transformative role. The mathematician first explored the computer-generated
picture with his pen. Then the independent line itself, no longer embedded in
the coordinate system, became a subject for mathematical investigation. In a
radical reversal of the Platonic dictum that discursive thinking (διάνοια) requires
no visible forms, the drawing became both a tool that helped Douady to
understand the computer-supported mathematics of complex dynamics and a mathematical
object to be analyzed by his διάνοια.
The three steps from donnée brute to drawing recall the
function of disegno in early modern
times, as employed, for example, by the artist, biographer, and theoretician
Giorgio Vasari in the mid-sixteenth century in a preliminary drawing for his
fresco The Studio of the Painter
(after 1561). There the viewer gazes into a pair of barrel-vaulted spaces
connected by a corridor and doors. On the right, female models are assembled in
front of a multi-breasted Diana Ephesia meant to embody the laws of nature.
They are about to step over into the left half of the picture, where a painter
is working at his easel, his gaze directed at additional posing models. The center
compartment contains a niche holding a male statue, identified by its
attributes and triple profile as a personification of disegno. Vasari here translated his theory of art, formulated in
the second edition of his Lives of the
Artists (1568), into a concrete pictorial allegory: As a draftsmanly step in
the development of an idea, disegno,
the basis and source of all arts, plays an intermediary role between nature and
art.
Douady’s conversion of the computer
picture into a curving line translated the classic use of disegno into a method for generating a mathematical theory. In a “corrective
overview,” the mathematician extracts a mathematical concept (concetto) from the “raw given” by means
of drawing. In both cases, it is only the drawing that makes possible the
process of visual transformation. As opposed to Vasari’s practice, however, the
end product of draftsmanly conceptualizing in mathematics is not a unique work
of art or the perfect image but rather a theoretical notion coagulated into a
drawn line, a “snapshot” of knowledge, which is fed into the thinking process
and used for further deliberations.
Once the telling lines had been
extracted from numerical mathematics, they could function as a way of stemming
the tide of serial similarities characteristic of images in this field of
study. The repetition of nearly identical images can be tiring. Douady relates
that he once almost fell asleep in a Paris café in 1982 when presented with a
series of similar pictures composed of mathematical colleague John Hubbard had placed
before him. Fresh from the printer on so-called continuous paper
perforated at the edges, they represented seemingly endless variations of Julia
sets, graphic depictions of rational functions produced by a dot-matrix
printer. Hubbard was unable to make out the significance of the pictures and
so had confronted his colleague with a whole stack of computer graphics. Douady
tells of the encounter in such a colorful way that it is worth quoting him
directly: “He asked me: ‘Can you recognize anything?’ I had nearly fallen
asleep, and could only see a succession of clouds that said nothing to me.
Suddenly I said to him: ‘There, an upside-down rabbit!’—‘Where?’—I took an orange
marker and traced it in the picture.” Douady had observed the succession
of similarities, and it was only the recognition of a shape and his drawing
that relieved him from boredom. The “rabbit” that Douady felt he could see in
the clouds of dots was a playful term for a specific quadratic Julia set, the
characteristics of which he had intently studied for years. It owes its name “fractal
rabbit” (lapin fractal) to what
appear to be ears extending from either side of the globular body. In his
draftsmanly retracing of the lapin,
the scientist was treating the digital form as if it were clay or rubber: The
rounded center section was taken apart and the “ears” were folded down along
the inner edges and placed in perfect alignment with the sides of the printed
pattern. Douady had turned the graphics into usable material for thought. In
the process of his trying to understand the data before him, the line derived
from the cloud of pixels came to seem like a plastic material, pliable and
ductile. While the drawing in the first example helped him to review the basic
structure, he now adapted the branching line of the Hubbard tree as if it were
made of wire and could be bent at will. What was of importance mathematically
was the way it converged in the center of the picture with the rabbit’s
sweeping semicircular arcs and the loops surrounding the image.
Despite its humor and subjective, anecdotal
character, Douady’s description of his discovery is of central importance in
the history of the representation and understanding of complex dynamics. What Douady
and Hubbard first discovered together in pictures they were later able to transform
into a valid mathematical proof. They gave the name “matings” (“accouplement fractal”) to their theorem
because the mating of two Julia sets of different classes of equations
completely fills the field of complex numbers that mathematicians refer to as
the Riemann sphere. Although the proof had to be worked out in an
analytical way based on formulas, Douady insists that it was precisely the act
of drawing that led to the mathematical insight: “And just at that moment,
quite suddenly, as soon as we had made the drawing, we understood what was going
on.”
In his handwritten chronology of
the events, Douady emphasized the moment of recognition with a drawing
interrupting the lines of text. Its shape, somewhere between an oval and a
heart with loops turned inward, gives the impression of an eye gazing out
between the lines. It can be regarded as a symbol for drawing as a thought
process in mathematics.
A number of Douady and Hubbard’s drawings
for the “matings” are known. One of their preferred methods in the early 1980s
was drawing on semitransparent foil. In figure 35, one can see how Douady traced
different curving lines in red, blue, green, and violet. The “rabbit” is
visible in red. The drawing is anchored in the transparent picture space by two
slightly curving vertical lines. Vigorously curving loops and loosely
distributed circles alternate with trial lines, some of which are struck
through with hatching. These are traces of mathematical exploration that
constitute a virtual allegory of the process of cognition, of its tentative searching,
calculating, making of mistaken assumptions, and experiencing of reversal and
reinforcement. Here, only by means of drawing, mathematical reflection could be
seen as both controlling the digital space and becoming independent of it: With
the use of foil as a transparent picture support, the visualized theory could
be literally liberated from the mass of raw data. This drawing of course was
not the proof, but it did provide evidence of what could later be confirmed.
This is the core notion of the second possibility presented in this essay: that
it is hand drawing that enables mathematical understanding in the realm of
chaos, where serial pictures form sequences of similarities.
Analog and digital techniques are
by no means mutually exclusive in fractal mathematics; they require and support
one another. The drawing is not replaced by computer graphics; it enhances them
and makes them intelligible. In fact, it is precisely owing to computer
graphics that the pencil has become the mathematician’s indispensable thinking
tool when extracting a theoretical concept from the mass of visual data.
Douady, a skilled draftsman,
produced a large number of other versions of his lapin. Made up of innumerable curves, curlicues, spirals, swirls,
and snaking lines, his lapins explore
geometrical relationships that can be proved mathematically, to be sure, but
cannot be calculated and depicted by any computer. The center composite figure,
labeled “the fat rabbit,” is composed of five larger almond shapes and small,
rounded buds. This basic figure is transposed into new configurations of serpentine
lines in the surrounding drawings. Douady’s imaginative figures, drawn with
great verve, show that research in this field of mathematics, first opened up
by computer graphics, involves working with pictures even when the computer
“blinds out,” the formulas having become too complex for digital
representation. At the time Douady drew them, the various curlicues he imagined
were as yet nonexistent in a mathematical sense inasmuch as their mathematical
properties were not accompanied by a rigorous proof. The proof would be the job
of someone else: What Douady’s drawing disclosed was a realm of possibility, a
potential mathematical scenario. Dan Sørensen, a doctoral student of
Douady’s, finally provided the proof of the scenario’s existence—and of the
efficacy of Douady’s loopy imagination—in his dissertation.
In Douady’s sketches, the backward S-curve
that the Bourbakists placed in the margins of their Éléments de Mathématique celebrated not only a return but an
ecstatic revival. They also came to be symbolic of the factual force of
drawings, for it was possible to prove mathematically the existence of the curlicue
dynamic. Douady can thus be identified not only as a second-generation Bourbakist,
but also as an exquisite Bourbakist draftsman.
Computer-graphic representations of
fractals, especially those based on combinations of polynomials, are
characterized by complex filigree structures around the edges. In order to
study the relative position of their individual components, mathematicians draw
“thought trees.” For example, as a combinatorial study, a drawing by
Jean-Christophe Yoccoz from 1983 represents a small portion of the Mandelbrot
set as mathematicians “see” it. In his drawing, the calculation becomes a
kind of hybrid between a tree and a molecular structure that fills the entire
sheet. A base function is seen at the bottom edge of the picture, a curving
line reminiscent of a small hill. Rising up out of it is a gnarly trunk with straight
and curved branches that sprout tiny twigs in every direction, an organic tangle
whose leaves or fruits are numbers. Each of the tree’s branches is the
expression of a calculating thought movement, and together they result in a
complex dynamic of their own: The upper right is dominated by rhythmic lines
curving to the left, repeating the movement of the exuberant formation in the upper
left. Douady explained the combinatory practice as “tree drawing,” which
represents one of the basic methods for working with polynomials and serves to tame
the structures visualized in computer graphics. Mathematical draftsmen
thus become engineers of a computer-generated observation room; in drawings,
digitally calculated abstract space is surveyed and virtually reconstructed.
Computer graphics may have been required to disclose these realms to
mathematics, but in scientific debate they had to be interpreted in drawings. Though
the computer helped access a realm of pure thought in which the world of
numbers might map itself seemingly unaided, drawing was essential for the return
of the eye, which in Mandelbrot’s view was primarily manifested in digitalized mathematics.
Drawings will always play an
indispensable role in mathematical thinking at the theoretical stage. Yet the
popular media are dominated by computer graphics that by and large mask the
process of coming to understanding through drawing. To the media public, the
presumably objective computer graphic is better at illustrating phenomena from
the exact sciences. But the shape of the graphic has most likely been designed
by a drawing hand, outside of public view, and only later presented in polished
form.
Of course, it is possible to think
in formulas—or, like Mandelbrot, in analogies— but arriving at understanding can
also be a matter of fiddling with pencil and paper, making drawings that lead
to a fruitful collaboration between the computer and the eye and thereby add to
mathematical knowledge.
Douady ended the article in which he
presented his theory of mathematical disegno
in 1982 (published, ironically, in the Séminare
N. Bourbaki) with a highly technical demand: Construction on the foundations
of the “chirurgy of dynamic systems” must continue. His wording can be
related to a humourous photomontage from his private archive, a spoof on a
French leftist poster of the French Communist Party (PCF) in which John Hubbard
and Adrien Douady have taken the places of the Gaullist politicians Jacques
Chirac and Charles Pasqua. Douady’s left hand rests on a three-dimensional drawing
of a Julia set, which appears to hover slightly above a table (where in the original
had been a map of France), while Hubbard is brandishing a butcher knife in his
right hand. They are gazing at the viewer with determined expressions. The two
mathematicians’ intention is spelled out in large letters next to their heads: Charcutage Polynomial (Polynomial
slaughter). This poster is a lively indication of the sparkling sense of humor
for which Adrien Douady was widely known and appreciated. But there is a deeper
meaning hidden in this joke: Had the two mathematicians not laid a hand and a
pencil on the “slaughterhouse of the Julia sets,” they would not have derived
any information from the raw digital shapes of visualized mathematics. In this
context, drawing became a mode of mathematical seeing: It was the linear that made
the non-linear visible.
Material for Thought:
Paper as a Threshold to the Mind; or, Paper Poiesis
A mathematician, on
the other hand, has no material to work with but ideas.
—Godfrey Harold Hardy
Anyway, I think with
my knee.
—Joseph Beuys
Experimental thinking, the cultural historian Thomas Macho
has explained, does not take place solely in the mind: “Intellectual experiments
are generally experiments on paper. We have to produce Flatland on our writing
tables in order to be able to give an account of it.” Macho was of course referring
to Edwin A. Abbott’s 1884 novel in which the author describes a world where geometrical
shapes play the main roles. Ninety years later, Otto E. Rössler’s way of using
pictures would provide an indication of how fundamentally important material evidence
can be to experimental thinking in mathematics. With Rössler, our concluding case
study is devoted to one of the most radically conceived and implemented collapsings
of the boundaries between the world of thinking and the world of things that I
encountered during my research in this field. We have already seen that in
their mathematical thinking, Benoît Mandelbrot and Adrien Douady employed
pictures in quite different ways. Rössler, in his work on the non-linear
dynamics of complex systems, utilizes physical representations in his thinking
in yet a third way. As with Douady, drawing is of utmost importance, but with Rössler,
its emphasis expands dramatically from a focus on the visual side to its material
qualities, which, in the manner of a feedback loop, directly influence the
properties of the mathematical objects under study.
Rössler, a biochemist by training,
is considered a pioneer in the field of “chaos theory” because of his
mathematical works from the mid 1970s. For the moment, the term “chaos”
has deliberately been placed in quotation marks, as it conveys something very
different to mathematicians than it does to laymen, for whom it is synonymous
with “confusion” or “lawlessness.” Chaotic systems form an order at the highest
level. Accordingly, a more precise term is “deterministic chaos.”
“Chaotic” means “not precisely predictable in the long term,” to be sure, but
by no means “arbitrary” or “unruly.” Scientists working in this field have
discovered forms of chaos as varied as they are precise. At the same time, they
have suggested making chaos to a certain extent controllable; pictures of chaos
are therefore always also “images of controllability.” The field of fractals is
linked to that of deterministic chaos. It can be thought of as a geometry of
chaos. Thus, if deterministic chaos describes a temporal complexity, fractals deal
with a spatial complexity.
In 1975, Rössler discovered an
“attractor,” a key concept in dynamics. Attractors can be described as
spatially complex geometric shapes. In simple terms, they illustrate all the
states a dynamic system can pass through over time. They are symbolic representations
of long-term predictions. If they are chaotic, they are called “strange.” The “Rössler
attractor” is one such “strange” shape. Rössler was deliberately searching
for a “simplification” of the chaotic model discovered in 1963 by the
meteorologist Edward Lorenz at MIT. The Rössler model is extremely
important in chaos research because it is reduced to the minimum requirements
necessary to exhibit chaotic behavior and therefore keeps analysis of the dynamic
system as simple as possible.
To explain how he first developed
this shape, Rössler drew spiral-like tracks on two sheets of paper. He then proceeded
to manipulate them in complex ways in three-dimensional space in an effort to
understand them. By studying them one above the other but at an angle, he
managed to “force in the paper” a figure whose structure approaches the
limit of what can be represented and imagined: an infinitely flat,
two-dimensional thing described by the movement of a one dimensional point that
nevertheless occupies a three-dimensional space. It was only the manipulation
of the two sheets of paper in various positions that gave him the sudden idea
that the desired shape could only be realized by a dimensional leap. Once the line
had arrived in the third dimension in its temporal development, it necessarily
had to be catapulted back into two dimensions. Since it could be continually
extended and retracted between the two sheets of paper arranged in space, it
was indeed possible to return it to the second dimension.
This account of his research
methods and the circumstances in which he made his discovery in December 1975
dates from some thirty years after the events. Looking back, Rössler confirmed
that it was in making the drawings and playing with them in space that he
understood what he was after analytically; his subsequent development of the
related formula—“the intrusion of a wedge of proof”—was merely a
mechanical act of realization. “[E]verything that one can imagine one can also
draw,” Rössler claimed, “and indeed much better, because in drawing one has
what is essential… . It always begins with the pictures.” To emphasize
this mechanism, Rössler noted a provocative triad that he had written on one of
his sketch sheets: “forcing— snapping—death bite” (“erzwingen— knacken—Tötungsbiß”). Retrospectively surprised by the
strong metaphors he had invented, he later commented that this “death bite”
would certainly be a “friendly” or “natural” one, as achieved, for example, by the
reflexes of a dog. It was as if the chaotic attractor had to be tamed in an
actual tour de force, or in a process resembling “catching the prey.” One could
dismiss Rössler’s description of his discovery through drawing as a fanciful
projection after the fact were it not that recently discovered sketches
document the process in detail. In scientific practice, preparatory sketches leading
to a discovery are generally trashed, hardly worth preserving once the thinking
process is finished. The preservation of this series of drawings is accordingly
a historic stroke of luck. Dating to the period between November 23 and
December 11, 1975, they provide eloquent testimony to Rössler’s habit of
thinking by drawing. One sees how he took up the basic shape of the Lorenz
model—two interconnected spirals— in the first drawing and then varied it step
by step. He stretched the shape into an oval, deformed it into wavy shapes, and
pressed it into the bottom corner of the coordinate system. Somewhat
offhandedly, he wrote, “soft watches (Dalí)” on another drawing. What was the
name of the famous Surrealist painter doing among Rössler’s working sketches?
One has to know that in chemistry there are cyclical reactions known as
“chemical clocks.” On the left in figure 40, one sees a “diagram clock” that Rössler
drew like an ear being struck by sound. Yet this equation does not produce a chaotic
reaction, for in order to trigger one, the materiality of the image from
chemistry—the clock—would need to be literally made “to melt,” as in the famous
Dalí painting The Persistence of Memory
(1931). This is explored in the other Rössler sketches here. Folding and
stretching had already been identified in mathematics as characteristics of
chaotic dynamics in the sixties; what was new was that by transferring the
material malleability of the stable “chemical clock,” a “chaos machine” might
be created. The idea that the material might be deformed was crucial to Rössler’s
subsequent discovery of the chaotic attractor through drawing. On the sheet
with the “soft watches,” there is also a sketch illustrating the step that
immediately preceded Rössler’s discovery. It pictures the loops, now layered one
behind the other in space, placed in a position to make possible the decisive step:
the dimensional leap that Rössler characterized as the switch between two
sheets of paper. In a small and seemingly unimposing sketch in the center of
the last sheet, one sees the dimensional leap taking place in the more darkly
hatched area between two broken lines. Once Rössler had become aware of the
ramifications of his discovery, he marked the small spiral drawing quite
clearly with a green arrow—a kind of ex post facto exclamation point extolling
the virtues of working out one’s thinking on paper.
It was only
after working out this series of simulations on paper that Rössler managed to
come up with the equations for his “handmade” chaos model and to visualize them
with the help of his analog computer. The crucial preliminary role of his “paper
models” and the relationship between the substance of his thinking and that
which his hands could touch was highlighted still further when in 1977 he
attempted to expand his model into the fourth dimension. To conceive of a new
dimension at all, his hands had to first experience it intuitively. His experiments
were now carried out with paper reshaped in space. Once again, a dimensional leap
was required in the mathematical model, one that he could only grasp by rolling
the paper. Rössler published drawings of this paper model in the Bulletin of Mathematical Biology.
Thanks to the physical and tactile aids in his research, he was able to
“pre-envision” what could only later be depicted in computer graphics with the
help of formulas and algorithms. The computer alone would not have sufficed. Rössler’s
colleagues greeted his procedure with amazed praise; as the mathematician Bruce
Stewart put it: “What was brilliant about what Otto Rössler did was that he
conceived of the shapes before the equations.”
Rössler’s
procedure can thus be seen as a radical expansion of Douady’s position. Not
only did the drawing trigger the sudden moment of understanding, the material substance—the
paper model—seemed to sculpt the supposedly immaterial digital image itself.
This opened up his thinking to the realm of possibility: “It is a mental experiment,
but it is always real things that one manipulates.” The rolling of the
paper and even the scratching of the pencil can be identified as “material
springboards to thought” or “thinking thresholds,” which influence the
structure of creative thought, whether in art or mathematics. Rössler was,
moreover, a sharp critic of digital pictures. Not only were they incapable of
achieving a return of the eye into
mathematics; strictly speaking, he insisted, they were not even “pictures”: “A
picture that is developed by way of algorithms is not truly a picture.”
Rössler describes the process of
arriving at understanding by handling paper or drawing on it as one that has
audible, even physical aspects: “If one makes a drawing on paper, or on two
sheets of paper, at some point it snaps into place, it clicks. One has to force
it, ‘adjust’ the shape so to speak. It then clicks audibly into place: It makes
your ears pop.” A mental image takes shape through the gesture of drawing:
It is a physical act just like any other physical experimentation, and as such
it leads to understanding. Rössler quotes the behaviorist Konrad Lorenz’s
well-known statement that “Thinking is handling things.” In his reports, Rössler
repeatedly relates how his own body becomes involved in his experiments. For
example, he tells of a “nose perception” that he claims occurred as he was
making his discovery. He had the feeling that a string kept winding around and
around his nose. The figure of the spiral attractor taking shape in his
imagination was virtually assaulting him. The researcher’s own body here
becomes a virtual “support” for the mathematical objects in defiance of the
Western belief that the physical plays no part in “objective research.”
There is, at times, an air of the
esoteric or even occult in Rössler’s descriptions of his working method, one
that is symptomatic of the intersubjectivity of mathematical picture worlds in
general. To mathematicians, pictures that lead to understanding are often difficult
to describe. The French mathematician Jacques Hadamard, who described his thinking
in images as sequences of “undefined specks,” provides a famous historical example.
In his proof that there are prime numbers greater than 11, his first step read:
“I consider all primes from 2 to 11, say 2, 3, 5, 7, 11.” To his thinking, this
was equivalent to saying: “I see a confused mass.” Then in a final step, the
closing solution: “I see a place somewhere between the confused mass and the
first point.” One has to imagine the intervening mental steps—that is to
say, the path from “confused mass” to the previous point—as moving images or a
dynamic sequences of pictures. It is not known whether or when Hadamard made
drawings. But his insight that mathematical thinking occurs in visual and not
always clearly definable images unites him with Rössler. In Rössler’s
case, however, one has to add that these “mental images” can no longer be
separated from the material world. As a process, thought itself becomes
material, and this materiality of thinking can be seen as a habit that does not
necessarily have to be conscious: “You just introduced the term haptic—that was a new formulation for
me, but you are correct. It is only new to me as a name, not as experience.”
In this, Rössler can be said to
agree with Friedrich Nietzsche that “our writing tools influence our thinking.”
Whereas Nietzsche loved to “think while writing,” to Rössler there is
something erotic about pictures: “Real pictures are always erotic pictures as
well. Topology or geometry is like looking from all sides as though in outer space.
That is all completely tactile. Without touching and without kneading it is
impossible to imagine. Materiality does matter: Essentially mathematics is
nothing more than pottery.”
One has to resist confusing Rössler’s
creative process as a mathematician with that of Prometheus, who as a skilled
sculptor is said to have fashioned mankind out of clay. Rössler’s approach is
more reminiscent of the “organic machines” designed in the mid-seventies by the
artist Joseph Beuys, in which materiality was basically perceived from the
point of view of its energy potential. In Honey
Pump in the Workplace, created by Beuys for documenta 6 in 1977, the artist
pumped honey, conceived of as a source of energy, through a system of plastic
tubing. With this “model of man,” Beuys designed a “technology … that is
first of all, in fact, an internal one, that is a conceptually creative one in
mankind himself.” Just as the circulating honey can be understood as the
circulation of blood of such an “organic machine,” Rössler’s rolling of
paper sheets was a way of designing chaos by thinking in material terms.
Prometheus molded man out of clay so as to breathe life into him and give him
the ability to reason. Rössler molded paper as a way of opening up to his
thinking new realms of possibility.
Epilogue: Interfaces
The mathematical mind has always seen itself confronted with
the problem of generating sense out of sequences of similarities. Here we have
been able to distinguish three basic ways of pictorial thinking by examining the
working materials of three different scientists. Benoît Mandelbrot elevated
analogy to a guiding principle and saw new pictures of the world taking shape
in oscillating similarities, while his islands pointed to the double-edged,
imaginative, and seductive power of the visual and identified drawing as means
of authentication. For Adrien Douady, the drawing of a line as a kind of mental
sword through the drift of dots in a series of similar computer graphics led to
a breakthrough in understanding. To Otto E. Rössler, the experimental
manipulation of things was equivalent to thinking—and all the images in the
various configurations of his sheets of paper were one and the same picture.
His research method attested to the dignity of the single picture as such.
Rolling his sheet of paper while pondering four-dimensional chaos, he was not
merely rolling paper. His thinking was rolling, or alternatively, he was
thinking the rolling, or perhaps even better, the rolling was thinking him. As
Maurice Merleau-Ponty was also aware, “One has to understand that things own
us, not we things.”
Creative processes
in art and the methods of the sciences both suggest that mental and physical
activities are interdependent and influence each other. The practice of “thinking
with one’s hand,” the subject of the present book and the related exhibition, coincides
with Beuys’s thinking “with my knee”—both are suggestive of a performative
model of thought. It is simply mistaken to assume that there is an autonomous mental
realm independent of the world of sense impressions, one in which ideas and concepts
can be pursued without reference to the material world—whether in the practice
of art or mathematics.
When trying
to categorize more closely the evidence—or
traces—that mathematical objects
leave behind in the world, as Jean-Pierre Changeux proposed at the very beginning
of this essay, it is only possible to conceive of them as interfaces between
the world of mathematical thought and visible reality. It is their very double
allegiance that gives them their potency and is the basis for their unique
dynamic: They do not dissipate somewhere between the two spheres but leave behind
traces, not only on paper but in thinking itself.
Jean-Pierre Changeux and Alain Connes, Conversations on Mind, Matter, and
Mathematics (Princeton: Princeton University Press, 1995). The original
title of their book, Matière à Pensée,
evokes the double meaning of “materials to think about” and “the materiality of
thought.” The present essay builds on previous of my publications, starting
with my dissertation, “Die Form des Chaos. Bild und Erkenntnis in der komplexen
Dynamik und der fraktalen Geometrie” (forthcoming, 2013, from Fink Publishers)
as well as “Form und Farbe digitaler Mathematik: Vom Zusammenspiel von
zeichnender Hand und Computer in fraktalen Bildwelten,” Bildwelten des Wissens, Kunsthistorisches Jahrbuch für Bildkritik
3 (2005): 18–31, and “‘I Look, Look, Look, and Play with Many Pictures,’ Zur
Bilderfrage in Benoît Mandelbrots Werk,” in Verwandte
Bilder. Die Fragen der Bildwissenschaft, ed. Ingeborg Reichle, Steffen
Siegel, and Achim Spelten (Berlin: Kadmos, 2007), 297–320.
Changeux and Connes, Conversations on Mind, Matter, and Mathematics, 44.
Italics in the original. While their publication contains some illustrations,
they do not go into particulars.
Benoît Mandelbrot, Fractals
and Chaos: The Mandelbrot Set and Beyond (New York: Springer, 2004), 34. Italics
in the original.
Their existence was known, nevertheless, as he mentions them
in Mandelbrot, “People and Events Behind the ‘Science of Fractal Images,’” in The Science of Fractal Images (New York:
Springer-Verlag, 1988), 8. One of the films was titled “Competing with the Lord
on the First Day of Creation.”
Benoît Mandelbrot, “Intermittent Turbulence and Fractal
Dimension: Kurtosis and the Spectral Exponent 5/3+B,” in Turbulence and Navier Stokes Equations (Orsay, 1975), ed. Roger
Temam (New York: Springer, 1976), 121.
H. G. Wells, The
Island of Dr. Moreau (Garden City, NY: Garden City Publishing Co., 1896);
see also R. M. P. Bozzetto and Russell Taylor, “Moreau’s Tragi-Farcical Island,”
Science Fiction Studies 20 (1993):
34–44. See also Günter M. Ziegler, “Countdown. Über Mathematik und Science
Fiction,” in Gegenworte. Hefte für den Disput
über Wissen. Bilder (in) der Wissenschaft 20 (2008): 73–82.
For this motif, see Horst Bredekamp, “Der Mensch als ‘zweiter
Gott.’ Motive der Wiederkehr eines kunsttheoretischen Topos im Zeitalter der
Bildsimulation,” in Bilder bewegen.
Von der Kunstkammer zum Endspiel, ed. Jörg Probst (Berlin: Wagenbach,
2007).
See Margarete Pratschke, “Charles and Ray Eames’ ‘Powers
of Ten’—Die künstlerische Bildfindung des Atoms zwischen spielerischem Entwurf
und wissenschaftlicher Affirmation,” in Atombilder.
Ikonografien des Atoms in Wissenschaft und Öffentlichkeit des 20. Jahrhunderts,
ed. Charlotte Bigg and Jochen Hennig (Göttingen: Wallstein-Verlag,
2009), 21–30.
Benoît Mandelbrot, interview by the author, Cambridge, MA,
April 2008.
Benoît Mandelbrot, The
Fractal Geometry of Nature (New York: Freeman, 1982), 21.
Magdolna Hargittai and István Hargittai, Candid Science Four (London: Imperial
College Press, 2004), 497. Other encomiums were similarly offered: As a commencement
speaker at Syracuse University, which awarded him an honorary doctorate in
Science and Engineering in 1986, he was lauded for “unraveling the secrets of
the universe”; when Boston University made him a Doctor of Science in 1987, it
was in recognition of “scientific, mathematical, and aesthetic insights by which
you have brought us closer to the character of the universe”; see http://www.math.yale.edu/mandelbrot
(accessed on December 10, 2011).
For photographic portraits of scientists and their
relation to popularization, see Daniel Jacobi and Bernard Schiele, “Scientific
Imagery and Popularized Imagery: Differences and Similarities in the
Photographic Portraits of Scientists,” Social
Studies of Science 19 (1989): 731–53. For the tradition of scientists’
portraits deriving from the nineteenth century, see Gabriele Werner, “Das Bild
vom Wissenschaftler—Wissenschaft im Bild. Zur Repräsentation von Wissen und
Autorität im Portrait am Ende des 19. Jahrhunderts,” kunsttexte.de 1 (2001): 1–11.
For his broken Dimension D—the “Hausdorff Dimension,” an
important feature of fractals—Mandelbrot adapted a definition formulated in
Felix Hausdorff’s “Dimension und äusseres Mass,” Mathematische Annalen 79 (1919): 157–79.
“LBLGRAPH” (or Labelgraph) was the name of a graphic
program at IBM; see Sigmund Handelman, “A High-Resolution Computer Graphics
System,” IBM Systems Journal 19
(1980): 361.
Benoît Mandelbrot, “People and Events Behind the ‘Science
of Fractal Images,’” in The Science of
Fractal Images (New York: Springer, 1988), 8–10. See Alain Fournier, Don
Fussell, and Loren Carpenter, “Computer Rendering of Stochastic Models,” Communications of the ACM 25 (1982):
371–84.
Benoît Mandelbrot, The
Fractal Geometry of Nature, 21.
See Benoît Mandelbrot, “Les inattendus des fractales,” pour la science 234 (1997): 10–11.
For a mathematical critique, see Steven G. Krantz,
“Fractal Geometry,” The Mathematical
Intelligencer 11 (1989): 12–16.
Gottfried Boehm, “Die Wiederkehr der Bilder,” in Was ist ein Bild?, ed. Gottfried Boehm
(Munich: Fink, 1994), 35. Boehm here refers in general to “electronic simulation
techniques.” For fundamental contributions to the critical art-historical
discussion, see Karl Clausberg, “Feigenbaum und Mandelbrot. Neue Symmetrien zwischen
Kunst und Naturwissenschaften,” Kunstforum
International 85 (1986): 86–93; Karl Clausberg, “Am Ende Kunstgeschichte?
Künstliche Wirklichkeiten aus dem Computer,” in Kunstgeschichte—aber wie? Zehn Themen und Beispiele, ed. Clemens
Fruh, Raphael Rosenberg, and Hans-Peter Rosinski (Berlin: Reimer, 1989),
259–93; and Horst Bredekamp, “Mimesis, grundlos,” Kunstforum International 114 (1991).
See the exemplary Jean Baudrillard, Die Agonie des Realen (Berlin: Merve,
1978); Jean-François Lyotard, La Condition
postmoderne: rapport sur le savoir (Paris: Les éditions de Minuit, 1979);
and Baudrillard, Subjekt und Objekt: fraktal.
Vortrag im Kunstmuseum Bern (Bern: Wabern, 1986). For a reaction to
fractals and chaos in the context of postmodernism, see Vilém Flusser,“Curie’s
Children: Vilém Flusser on an Unspeakable Future,” Artforum 19 (1990): 22–23; Vivian Sobchack, “A Theory of
Everything: Meditations on Total Chaos,” Artforum
19 (1990): 48–55; Michael Speaks, “Chaos, Simulation, and Corporate
Culture,” Mississippi Review
17 (1989): 159–76; Richard Wright, “Art and Science in
Chaos: Contested Readings of Scientific Visualizations,” in FutureNatural: Nature, Science, Culture,
ed. George Robertson (London: Routledge, 1996); Horst Bredekamp, “Der
simulierte Benjamin. Mittelalterliche Bemerkungen
zu seiner Aktualität,” in Frankfurter Schule
und Kunstgeschichte, ed. Andreas Berndt, et al (Berlin: Reimer, 1992),
117–40; and Horst Bredekamp, “Das Bild als Leitbild. Gedanken zur überwindung
des Anikonismus,” in LogIcons: Bilder
zwischen Theorie und Anschauung (Berlin: Sigma, 1997), 233–36.
This information is based on a conversation with
Aliette Mandelbrot on December 4, 2011, in Cambridge, MA.
Benoît Mandelbrot, “Fractals and the Rebirth of
Experimental Mathematics,” in Fractals
for the Classroom: Part One: Introduction to Fractals and Chaos, ed. Hartmut
Jürgens, Heinz-Otto Peitgen, and Dietmar Saupe (New York: Springer, 1992), 4.
Georg Cantor, quoted from Jean-Luc Chabert, “Un demi-siècle
de fractales: 1870–1920,” Historia
Mathematica 17 (1990): 348.
Charles Hermite, letter from May 20, 1893, in Benjamin
Baillaud and Henry Bourget, eds., Correspondance
d’Hermite et de Stieltjes, vol. 2 (Paris: Gauthier Villars, 1905), 318.
See Klaus Thomas Volkert, Die Krise der Anschauung: Eine Studie zu formalin und heuristischen Verfahren
in der Mathematik seit 1850 (Göttingen: Vandenhoeck & Ruprecht, 1986);
Herbert Mehrtens, Moderne—Sprache—Mathematik:
Eine Geschichte des Streits um die Grundlagen der Disziplin und des Subjekts formaler
Systeme (Frankfurt am Main: Suhrkamp, 1990); and Bettina Heintz, Die Herrschaft der Regel. Zur Grundlagengeschichte
des Computers (Frankfurt am Main: Campus Verlag, 1993).
Paul Du Bois-Reymond, “Versuch einer Classification der
willkürlichen Functionen reeller Argumente nach ihren Aenderungen in den
kleinsten Intervallen,” Journal für die reine
und angewandte Mathematik 79 (1875): 29 note.
Henri Poincaré, Science
et Méthode (Paris: Flammarion, 1908), 132.
Benoît Mandelbrot, “Des monstres de Cantor et de Peano à
la géométrie fractale de la nature,” in Jean A. Dieudonné et al., eds., Penser les mathématiques (Paris: Seuil, 1982),
233; for the aspect of “beauty,” see also Mandelbrot, “Les fractals, les monstres
et la beauté,” Le débat: histoire,
politique, société 24 (1983): 54–72. For the development of the terminology
of the “monsters,” see Klaus Thomas Volkert,
Die Krise der Anschauung, 99 ff.
Angela Fischel, Natur
im Bild: Zeichnung und Naturerkenntnis bei Conrad Gessner und Ulisse Aldrovandi
(Berlin: Mann, 2009), 118.
Georg Cantor, quoted from Chabert, “Un demi-siècle de
fractales: 1870–1920,” 348.
Mandelbrot, “Des monstres de Cantor et de Peano à la
géométrie
fractale de la nature,” 226.
Bernhard Siegert, Passage
des Digitalen. Zeichenpraktiken der neuzeitlichen Wissenschaften 1500–1900 (Berlin:
Brinkmann & Bose, 2003), 313, 318.
See, for example, David Hilbert, “Über die stetige
Abbildung einer Linie auf ein Flächenstück,” Mathematische Annalen 38 (1891): 459–60.
See Volkert, Die
Krise der Anschauung, 90.
Hermann Weyl, “Über die neue Grundlagenkrise der
Mathematik,” in Hermann Weyl. Gesammelte
Abhandlungen, ed. Komaravolu Chandrasekharan, vol. 2 (Berlin: Springer,
1968), 143.
See David Hilbert, Grundlagen
der Geometrie. Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals
in Göttingen (Leipzig: B. G. Teuber, 1899), 459–60.
For Hilbert’s radical axiomatic program, see Herbert,
Mehrtens, Moderne—Sprache—Mathematik,
108–42.
Ibid., 113.
See Heintz, Die
Herrschaft der Regel. Zur Grundlagengeschichte des Computers, 16–34. See
also Albert Einstein’s corresponding description of the axiomatic method in
Albert Einstein, Geometrie und Erfahrung.
Erweiterte Fassung des Festvortrages gehalten an der Preußischen Akademie der
Wissenschaften zu Berlin am 27. Januar 1921 (Berlin: Springer, 1921).
Konrad Knopp, Mathematik
und Kultur (Berlin: W. de Gruyter, 1985), 22.
See Paul R. Halmos, “Nicolas Bourbaki,” Scientific American 196 (1957): 89; also
Jean A. Dieudonn., “The Work of Nicholas Bourbaki,” The American Mathematical Monthly 77 (1970): 134 note. The name of
the general concerned was Charles Denis Bourbaki.
Nicolas Bourbaki, “Die Architektur der Mathematik,” in Mathematiker über Mathematik, ed.
Michael Otte (Berlin: Springer, 1974), 158.
Ibid., 144.
According to Pierre Cartier, quoted in Marjorie Senechal,
“The Continuing Silence of Bourbaki: An Interview with Pierre Cartier,” The Mathematical Intelligencer 20 (1998):
27.
Bourbaki, “Die Architektur der Mathematik,” 151.
Armand Borel, Mathematik:
Kunst und Wissenschaft (Munich: Die Stiftung, 1982), 33.
Bourbaki, “Die Architektur der Mathematik,” 158 ff.
See Maurice Mashaal, Bourbaki. Une société secrète
de mathématiciens (Paris: Belin, Pour la science, 2002), 56.
Horst Bredekamp, “Die Zeichnende Denkkraft. Überlegungen
zur Bildkunst der Naturwissenschaften,” in Einbildungen,
ed. Jörg Huber (Zurich: Springer, 2005), 163–65.
For the figura serpentinata
of the sixteenth century and its art-theoretical reinterpretation by Hogarth,
see Peter Gerlach, “Zur zeichnerischen Simulation von Natur und natürlicher
Lebendigkeit,” Zeitschrift für Ästhetik
und Allgemeine Kunstwissenschaft 34 (1989): 243–79.
See Régine Bonnefoit, Die Linientheorien von Paul Klee (Petersberg: Imhof, 2009), 9,
43–47. Graphical and theoretical investigations of the expressivity of lines are
no exception in the early twentieth century (cf. psychology, philosophy, graphology,
choreography). For a discussion of the serpentine line as a time-transcending symbol
of visual thinking and the movements of the mind, see Horst Bredekamp, “Die
Unüberschreitbarkeit der Schlangenlinie,” in Minimal concept: zeichenhafte Sprachen im Raum, ed. Christian
Schneegass (Dresden: Verlag der Kunst, 2001), 205–208.
David Hilbert, “Mathematische Probleme,” in Die Hilbertschen Probleme, ed. P. S. Alexandrov
(Leipzig: Akademische Verlagsges, Geest & Portig, 1983), 27.
In this spirit, the Old Testament refers to the symbol of
a copper snake, looking at which would save the life of the snake-bitten (cf.
Numbers 21:9). In this narrative, the mimetic reproduction of the threatening
phenomenon was credited with the power to defeat itself.
See Allyn Jackson, “Comme Appelé du Néant—As If
Summoned from the Void: The Life of Alexandre Grothendieck, II,” Notices of the AMS 51 (2004): 1199. He
called his “picture theory” “dessins d’enfants,” with pictures reminiscent of
children’s line drawings becoming the objects of investigation.
Benoît Mandelbrot, see interview in this volume, p.
155.
Mandelbrot, interview, April 2008.
Benoît Mandelbrot, see interview in this volume, p. 154
Benoît Mandelbrot, Fractals
and Chaos: The Mandelbrot Set and Beyond (New York: Springer, 2004), 3.
Benoît Mandelbrot, see interview in this volume, p.
154.
Ibid.
Benoît Mandelbrot, “How Long Is the Coast of Britain?
Statistical Self-Similarity and Fractional Dimension,” Science 156 (1967): 636–38.
Lewis Fry Richardson, “The Problem of Contiguity: An
Appendix to Statistics of Deadly Quarrels,” in General Systems: Yearbook of the Society for the Advancement of General
Systems Theory 6 (1961): 168. Like Mandelbrot, Richardson was an interdisciplinary
border crosser, involved in meteorology, mathematics, geophysics, statistics,
and even peace studies.
Quotes from Mandelbrot, “Les inattendus des fractales,”
10–11, and Mandelbrot quoted from Monte Davis, “Profile of Benoît B.
Mandelbrot,” Omni Magazine (February
5, 1984): 4.
Peter Glaser, “Das Kolumbus-Gefühl. Entdeckungen in
einer virtuellen Welt,” in Computerkultur
oder The Beauty of Bit & Byte, ed. Michael Weisser (Bremen: TMS, 1989),
19–40. Comparable descriptions—“a new world to conquer”—are found in connection
with microscopy; see Stefan Ditzen, “Der Satyr auf dem Larvenrücken. Zum Verhältnis
von instrumentellem Sehen und Bildtraditionen,” in Konstruierte Sichtbarkeiten. Wissenschafts- und Technikbilder seit der Frühen
Neuzeit, ed. Martina Hessler (Munich: W. Fink, 2006).
Benoît Mandelbrot, Les
objets fractals: forme, hazard et dimension (Paris: Flammarion, 1975), 116.
Benoît Mandelbrot, “On the Geometry of Homogeneous
Turbulence, with Stress on the Fractal Dimension of the Iso-Surfaces of
Scalars,” Journal of Fluid Mechanics 72
(1975): 405.
For chance methods in the creation of twentieth-century
art, see Bernhard Holeczek and Lida von Mengden, eds., Zufall als Prinzip: Spielwelt, Methode und System in der Kunst des 20.
Jahrhunderts, exh. cat. (Ludwigshafen am Rhein: Wilhelm-Hack-Museum, 1992).
For chance as a principle of the artistic avant-garde, see Christian Janecke, Kunst und Zufall. Analyse und Bedeutung
(Nuremberg: Verlag für Moderne Kunst, 1995); Horst Bredekamp, “John Cage and
the Principle of Chance,” in Music and
the Aesthetics of Modernity, ed. Karol Berger and Anthony Newcomb
(Cambridge, MA: Harvard University Press, 2005), 99–107. Herbert Molderings, Duchamp and the Aesthetics of Chance: Art as
Experiment (New York: Columbia University Press, 2010). For chance pictures
as “picture games of nature,” see Horst Bredekamp, Theorie des Bildakts: Über das Lebensrecht des Bildes (Frankfurt am
Main: Suhrkamp, 2010), 317–23.
Giorgio Vasari, Vasari
on Technique, trans. Louisa S. Maclehose (London: Dent, 1907), 212. In
later centuries as well, the relationship between blot and picture existed at
the interface between art and science, serving as a field for experimentation
for the testing and expansion of concepts like truth and objectivity. See
Friedrich Weltzien, Fleck, das Bild der Selbsttätigkeit: Justinus
Kerner und die Klecksografie als Experimentelle Bildpraxis zwischen Ästhetik
und Naturwissenschaft (Göttingen: Vandenhoeck & Ruprecht, 2011). In
relation to Mandelbrot’s interpretation of specks, another cultural technique comes
to mind, this time at the interface between psychosomatics and hermeneutics:
Rorschach blots, the technique developed by Hermann Rorschach in which ink
blots are used as a diagnostic tool for the psychological testing of
personality; see Hermann Rorschach, Psychodiagnostik.
Methodik und Ergebnisse eines wahrnehmunsdiagnostischen Experiments
(Deutenlassen von Zufallsformen), ed. Walter Morgenthaler (Bern: Hans Huber,
1972). According to Peter Galison, Rorschach tests became a “symbolic
technology” that made earlier psychological and philosophical debates about the
distinction between “seeing” and “seeing as” superfluous; see Peter Galison, “Image
of Self,” in Things That Talk: Object
Lessons from Art and Science, ed. Lorraine Daston (New York: Zone Books,
2004), 276, 291–93.
Vasari, Vasari on
Technique, 212.
Alexander Cozens, “An Essay to Facilitate the Invention
of Landskips,” quoted in Werner Busch, Das
sentimentalische Bild: die Krise der Kunst im 18. Jahrhundert und die Geburt
der Moderne (Munich: Beck, 1993), 337.
Alexander Cozens, “New Method of Assisting the
Invention in Drawing Original Compositions of Landscape,” quoted from Busch, Das sentimentalische Bild, 348. Cozens
thought of his manual as an expansion of the method suggested by Leonardo da
Vinci in his Treatise on Painting
through which forms encountered by chance in nature are used as sources of inspiration
for motifs in painting. In contrast to Leonardo, Cozens preferred to produce
the forms deliberately, “making those imperfect Forms on Purpose, and with some
degree of Design, which otherwise must be catch’d from Accidents”; Cozens, “An
Essay to Facilitate the Invention of Landskips,” 1, quoted from Kim Sloan, “A
New Chronology for Alexander Cozens Part II: 1759–86,” The Burlington Magazine 127 (June 1985): 356, illus. 7.
Busch, Das
sentimentalische Bild, 346.
Holeczek and von Mengden, eds., Zufall als Prinzip, 17.
Busch, Das
sentimentalische Bild, 348.
Hans-Jörg Rheinberger, Experimentalsysteme und epistemische Dinge: eine Geschichte der
Proteinsynthese im Reagenzglas (Frankfurt am Main: Suhrkamp,
2006), 27, 128. This corresponds to Rheinberger’s description
of the peculiarity of an “epistemic thing”; see ibid., 27–34.
Benoît Mandelbrot, “New Methods in Statistical
Economics,” The Journal of Political
Economy 71 (1963): 435.
William Feller, An
Introduction to Probability Theory and Its Applications, vol. 1 (New York:
Wiley, 1957), 83–85. This image only appears in the 2nd edition.
Mandelbrot, “New Methods in Statistical Economics,”435.
Mandelbrot, Les objets
fractals, 55; Mandelbrot, Fractals,
24; Mandelbrot, The Fractal Geometry of Nature,
257. In economics and signal transmission: J. M. Berger and Benoît Mandelbrot,
“A New Model for Error Clustering in Telephone Circuits,” IBM Journal of Research and Development 7 (1963): 225; Mandelbrot, “Self-Similar
Error Clusters in Communication Systems and the Concept of Conditional
Stationarity,” IEEE Transaction on
Communications Technology 13 (1965):
73. Further, in relation to Louis Bachelier, in Mandelbrot,
“Le syndrome de la variance infinie et ses rapports avec la discontinuité des
prix,” Economie Appliquée 26 (1973):
325.
See Edward R. Tufte, The Visual Display of Quantitative Information (Cheshire, CT:
Graphics Press, 2002), 32. See also Howard Wainer, “Graphical Visions from William
Playfair to John Tukey,” Statistical
Science 5 (August 1990): 340–46; and A. D. Biderman, “The Playfair Enigma:
The Development of the Schematic Representation of Statistics,” Information Design 6 (1990): 3–25.
In the second half of the eighteenth century, geology developed
as a scientific discipline with its own pictorial tradition; see Martin
Rudwick, “The Emergence of Visual Language for Geological Science 1760–1840,” History of Science 14 (1976): 149–95;
also Rudwick, Bursting the Limits of
Time: The Reconstruction of Geohistory in the Age of Revolution (Chicago:
University of Chicago Press, 2005).
One of the chapters in Ruskin’s major work, Modern Painters, from 1856, was titled
“The Sculpture of Mountains.” Ruskin was influenced by Charles Lyell, the founder
of modern geology, who in the years 1830–33 had questioned the notion of
biblical creation with his publication of Principles
of Geology; see Moritz Wullen, ed., Natur
als Vision: Meisterwerke der englischen Präraffaeliten (Berlin: SMB-DuMont,
2004), 43, 54. Like Mandelbrot later, Ruskin was of the opinion that mountains
are built up scale-invariantly and that therefore the structure of an entire
mountain is recognizable in the smallest rocks, a view he found expressed in
similar fashion in the “nature vision” paintings of the Pre-Raphaelites, whom
he admired and promoted. Ruskin’s thesis claimed that “the laws of geological
processes … can be visualized in a single piece of rock just as in an
entire mountain”; ibid., 46.
Compare the reception of the Feynman Diagrams, adapted
to visual conventions. See David Kaiser, “Stick-Figure Realism: Conventions,
Reifications, and the Persistence of Feynman Diagrams, 1948–1964,” Representations 70 (2000): 49–86.
According to the art historian Gottfried Boehm,
scientific pictures are deictic,
which means that their chief function is “to show.” Given their instrumental function,
such aesthetic criteria as wealth of allusion, metaphorical quality, visual
density, and self-reference hardly come into play; see Gottfried Boehm,“Zwischen
Auge und Hand. Bilder als Instrumente der Erkenntnis,” in Mit dem Auge denken. Strategien der Sichtbarmachung in
wissenschaftlichen und virtuellen Welten, ed. Bettina Heintz and Jörg Huber
(Vienna: Springer, 2001), 53.
Margit Rosen, “Die Maschinen sind angekommen. Die
[Neuen] Tendenzen visuelle Forschung und Computer,” bit international. [Nove]
tendencije—Computer und visuelle Forschung. Zagreb 1961–1973, ed. Margit
Rosen and Peter Weibel (Graz: Neue Galerie, 2007), 39.
See Karin Gludovatz, “Malerische Worte. Die Künstlersignatur
als Schrift-Bild,” in Schrift:
Kulturtechnik zwischen Auge, Hand und Maschine, ed. Gernot Grube, Werner
Kogge, and Sybille Krämer (Munich: W. Fink, 2005), 313. For signatures in the
sciences, see Alfred Nordmann, “Vor-Schrift—Signaturen der
Visualisierungskunst,” in Ästhetik der Wissenschaft. Interdisziplinärer Diskurs über das
Gestalten und Darstellen von Wissen, ed. Wolfgang Krohn (Hamburg: F.
Meiner,
2006).
Davis, “Profile of Benoît B. Mandelbrot,” 7. Capitals
in the original.
For a productive reflection on problems having to do
with pictures in art and science, see Peter Geimer, Bilder aus Versehen: eine Geschichte fotografischer Erscheinungen
(Hamburg: Philo Fine Arts, 2010), 294. Geimer quotes Gottfried Boehm’s warning:
“The maxims of a mere ‘expansion’—of art, but of science as well—only lead to
blurred simplifications”; Boehm, “Zwischen Auge und Hand,” 52.
Mandelbrot, interview, April 2008.
See Gregory F. Lawler, Oded Schramm, and Wendelin
Werner, “The Dimension of the Planar Brownian Frontier is 4/3,” in Citebase,
http://arxiv.org/abs/ math/0010165v2.
Gregory F. Lawler, e-mail to the author, February 2, 2012.
Benoît Mandelbrot, “Fractals and the Geometry of
Nature,” in 1981 Yearbook of Science and
the Future (Chicago: Encyclopedia Britannica, Inc., 1980), 168.
Mandelbrot, Les objets
fractals, 65. Mandelbrot compares the straight lines in the text with Baron
Haussmann’s redesign of Paris streets in the nineteenth century.
“En effet, il se propose d’établir qu’il existe entre tous
ces objets de profondes analogies, se situant à un niveau intuitif qui mérite
(malgré sa nouveauté) d’être considéré comme élémentaire.”
For an introduction to picture arrangements as “picture
tableaus,” see Margarete Pratschke, “Bildanordnungen,” in Das Technische Bild. Kompendium zu einer Stilgeschichte
wissenschaftlicher Bilder, ed. Horst Bredekamp, Birgit Schneider, and Vera
Dünkel (Berlin: Akademie, 2008), 116–19. For precursors of analogical thinking
about pictures in the Middle Ages—again with a cosmological dimension see
Wolfgang Kemp, “Mittelalterliche Bildsysteme,” Marburger Jahrbuch für Kunstwissenschaft 22 (1989). Analogy
formation in the twentieth century has not been adequately studied. However,
Barbara Stafford has presented an attempt; she accuses contemporary culture of
impoverished analogical thinking insofar as it places greater emphasis on
making distinctions than on recognizing similarities; Barbara Maria Stafford, Visual Analogy: Consciousness as the Art of
Connecting (Cambridge, MA: MIT Press, 1999), 32.
Mandelbrot, interview, April 2008.
To the square of a number a one is added, the result squared,
and again a one added, and so on, theoretically ad infinitum. Only numbers from
this procedure that do not add up to an infinite value belong by definition to
the set.
For a broader image-theoretical discussion of islands as
mediums of separation and deviation and a draft of “islands as cultural
technique,” see Gloria Meynen, “Die Insel als Kulturtechnik (Ein Entwurf),” Zeitschrift für Medienwissenschaft 2
(2010): 79–91.
The term “polynomial” is a compound term for the type
of equations under investigation here, with multiple variables and constants of
a finite length. For the foundational works for this field, see Gaston Julia,
“Mémoire sur l’itération des fonctions rationelles,” Journal de Math. pures et appliquées 8.1 (1918): 47–245; and Pierre
Fatou, “Sur les équations fonctionelles,” Bulletin
de la Soc. Math. Fr. 47 (1919): 161–271.
See Benoît Mandelbrot in A. Barcellos, “Interview of
B. B. Mandelbrot,” in Mathematical People
(Boston: Birkhäuser, 1984), 211.
See John Hamal Hubbard, preface to The Mandelbrot Set: Theme and Variations,
ed. Tan Lei (Cambridge: Cambridge University Press, 2000), xiii–xx.
Ibid., xv.
Benoît Mandelbrot, “Two Nonquadratic Rational Maps
Devised from Weierstrass Doubling Formulas,” in Mandelbrot, Fractals and Chaos: The Mandelbrot Set and
Beyond (New York: Springer, 2004), 157.
Ibid.
Ibid.
Ibid., 161.
This phenomenon was described by the mathematician
Samuel Lattès in 1918; see ibid., 139.
Ibid., 159.
Ibid., 169.
Ibid., 14 and 23.
For image interference and phantom images in a larger
context, see Peter Geimer, “Was ist kein Bild? Zur ‘Störung der Verweisung,’”
in Ordnungen der Sichtbarkeit. Fotografie
in Wissenschaft, Kunst und Technologie, ed. Peter Geimer (Frankfurt am
Main: Suhrkamp, 2002), 313–41.
Mandelbrot, Fractals
and Chaos, 14; and Mandelbrot, interview, April 2008.
It is interesting to note that Robert Brooks and Peter
Matelski had presented their own picture of the Mandelbrot set at a mathematics
conference the year before, though it was not published until three years later,
together with the rest of the conference papers; see Brooks, Robert, and J.
Peter Matelski, “The dynamics of 2-generator subgrounds of PSL (2, C),” in Riemann Surfaces and Related Topics: Proceedings
of the 1978 Stony Brook Conference, published in the Annals of Mathematics Studies 97, ed. Irwin Kra and Bernard Maski (Princeton:
Princeton University Press, 1981), 71. In their ASCII version of the set, the
bulbous shape is made up of individual little stars, with a lance-like
extension on the left side. Brooks and Matelski were in this sense “discoverers
without a discovery,” for theirs was the first picture of such a set presented
to the public, though they failed to recognize its self-similarity and
complexity because the surrounding specks were too small to be seen with the
visualization technology they used.
Mandelbrot, Fractals
and Chaos, 14.
See Mandelbrot, Les
objets fractals, 112–17.
See Nina Samuel, “‘Do Not Clean Off the Dust Specks.
They Are Real.’ Über gestörte, verschmutzte und verborgene Computerbilder,” in Prekäre Bilder, ed. Robert Suter and
Thorsten Bothe (Munich: Wilhelm Fink, 2009), 19–47. I place Mandelbrot’s
proscriptions against cleaning in the context of various other ways in which computer
pictures have been interfered with or become dysfunctional.
See Senechal, “The Continuing Silence of Bourbaki,” 26.
Benoît Mandelbrot, “Fractal Aspects of the Iteration of
z → λz (1-z) for complex λ and z,” Nonlinear
Dynamics: Annals of the New York Academy of Sciences 357 (New York: New
York Academy of Sciences, 1980), 250.
Mandelbrot, Fractals
and Chaos, 22.
See ibid., 37.
Adrien Douady and John Hamal Hubbard, “Itération
des polynômes
quadratiques complexes,” Comptes rendus
(Paris) 294 (1982): 123: “Ces îlots sont en fait rattachés par
des filaments qui échappent à l’ordinateur.”
Adrien Douady, telephone interview by the author, January
7, 2005.
For a discussion of the “two planes“ of the digital
image see Frieder Nake, “Das doppelte Bild,” Bildwelten des Wissens, Kunsthistorisches Jahrbuch für Bildkritik
3 (2005): 40–50.
Mandelbrot, The
Fractal Geometry of Nature, text for color plate C1.
Thomas Schlich, “Repräsentationen von
Krankheitserregern. Wie Robert Koch Bakterien als Krankheitsursache dargestellt
hat,” in eds. Hans-Jörg Rheinberger,
Michael Hagner, and B. Wahrig-Schmidt, Räume des Wissens. Repräsentation, Codierung, Spur (Berlin: Akademie,
1997), 170.
See Benoît Mandelbrot, “Fractals and the Rebirth of
Iteration Theory,” in The Beauty of
Fractals: Images of Complex Dynamical Systems, ed. Heinz-Otto Peitgen and
Peter H. Richter (Berlin: Springer, 1986), 157; Mandelbrot, Fractals and Chaos, 19.
Ibid., 17.
Edward Hill, The
Language of Drawing (Englewood Cliffs, NJ: Prentice-Hall, 1966), 25.
Douady, telephone interview. For the suppression of geometric
drawings from official discourse in the mathematical sciences, see Peter
Galison, “The Suppressed Drawing: Paul Dirac’s Hidden Geometry,” Representations 72 (2000): 145–66.
Douady, telephone interview.
The different variants of the “Hubbard tree” were
discovered by John Hubbard in 1980; see Hubbard, preface to The Mandelbrot Set: Theme and Variations,
xvi.
Plato, Republic,
in The Collected Dialogues of Plato,
ed. Edith Hamilton and Huntington Cairns, trans. Paul Shorey (New York:
Pantheon, 1989), 744–47.
This is based on the interpretation by Wolfgang Kemp
in “Disegno. Beiträge zur Geschichte des Begriffs zwischen 1547 und
1607,” Marburger Jahrbuch für Kunstwissenschaft
22 (1989): 121–34.
Ibid., 230.
“Nuages de points tracés à l’ordinateur,” in a letter to
the author, December 20, 2004. John Hubbard had some forty variations of such
“pictures of dot clouds.”
Douady, telephone interview, and letter to author,
December 20, 2004. The pictures showed “polynomial-like mappings” following a
method that Douady had explained to Hubbard.
Ibid.
This name became a permanent term in mathematical
literature. Douady also produced a didactic film on the “Dynamics of the
Rabbit”; cf. Adrien Douady, Dan
Sørensen, and François Tisseyre, La dynamique du lapin (Paris: Atelier Ecoutez voir, 1996).
Adrien Douady, “Systèmes dynamiques holomorphes,” Séminaire N. Bourbaki 599 (1982): 58–60.
For the Riemann sphere, which is thought of as a globe, see Hartmut Jürgens,
Heinz-Otto Peitgen, and Dietmar Saupe, Chaos:
Bausteine der Ordnung (Berlin: Springer, 1994), 361.
Douady, telephone interview.
For an art historical discussion of serpentines, see notes
49–51.
“Douady a dessiné les évolutions possibles des
formes,” Arnaud Chéritat, interview by author, Toulouse, February 8, 2008.
Dan Erik Krarup Sørensen, “Infinitely Renormalizable Quadratic
Polynomials, with Non-Locally Connected Julia Sets,” The Journal of Geometric Analysis 10 (2000).
Jean-Christophe Yoccoz (born 1957), French mathematician
to whom the Fields Medal was awarded in 1994 for his work on dynamical systems.
“You have to study it like a tree,” Douady, telephone
interview by author, January 7, 2005.
Hilbert, “Mathematische Probleme,” 27.
For another example of drawing as necessary for
achieving mathematical understanding in this specific field, see Nina Samuel,
“Beredte Skizzen. Chaos und das zerbrochene Ei von Yoshisuke Ueda,” Bildwelten des Wissens, Kunsthistorisches Jahrbuch
für Bildkritik 7, no. 2 (2010): 83–89.
Douady, “Systèmes dynamiques holomorphes,” 62.
G. H. Hardy, “A Mathematician’s Apology,” in The World of Mathematics, Vol. 4, ed.
James Roy Newman (New York: Simon and Schuster, 1956), 2027.
Joseph Beuys, Zeichnungen
1947–59 I. Gespräch zwischen Joseph Beuys und Hagen Lieberknecht. Geschrieben von
Joseph Beuys (Cologne: Schirmer, 1972), 17. Beuys came up with many
variations on this sentence over the course of his life.
Thomas Macho, “Das Rätsel der vierten Dimension,” in Science & Fiction. Über Gedankenexperimente
in Wissenschaft, Philosophie und Literatur, ed. Thomas
Macho and Annette Wunschel (Frankfurt am Main:
Fischer, 2004), 77.
See Otto E. Rössler, “Chaos, Hyperchaos, and the
Double-Perspective,” in eds. Ralph Abraham and Yoshisuke Ueda, The Chaos Avant-Garde: Memories of the Early
Days of Chaos Theory (Singapore: World Scientific, 2000), 209–19. For a
discussion of his place in the history
of chaos theory, see Christophe Letellier and Valérie Messager, “Influences on
Otto E. Rössler’s Earliest Paper On Chaos,” International
Journal of Bifurcation and Chaos 20 (2010): 1–32.
For a historical overview with personal reports and important
texts by “pioneers” in the field, see Abraham and Ueda, The Chaos Avant-Garde. For a further historiography see David Aubin
and Amy Dahan Dalmedico, “Writing the History of Dynamical Systems and Chaos: Longue
Durèe
and Revolution, Disciplines and Cultures,” Historia
Mathematica 29 (2002): 271–339.
For the first publication of the Rössler attractor see
Otto E. Rössler, “Chaotic Behavior in Simple Reaction Systems,” Zeitschrift für Naturforschung 31 (1976):
259–64.
See Edward Lorenz at MIT in 1963; see Edward N.
Lorenz, “Deterministic Nonperiodic Flow,” Journal
of the Atmospheric Sciences 20 (1963): 130–41. Lorenz was a meteorologist
and pioneer of chaos theory. He also coined the now-popular term “butterfly
effect.”
Otto E. Rössler, interview by author, Tübingen, Germany,
June 1, 2007.
Ibid.
Ibid.
His goal was the transformation of Lorenz’s model as a
biochemical reaction system. The typical
chemical-reaction diagrams with which he had been working for years served him
as a starting point, and can be seen at the top of fig. 40.
Rössler thereby transferred one of the basic
characteristics of chaotic dynamics onto his chemical clocks: folding and
stretching topology; see Steve Smale, “Differentiable Dynamical Systems,” Bulletin of the American Mathematical
Society 73 (1967): 747–817.
In this film, produced a year later (1976), he can be
seen operating the Dornier DO 240 analog computer that he used intensively from
1970 to 1979. See Letellier and Valérie Messager, “Influences,” 4.
In order to arrive at a new understanding, his
thinking had to be done “together with the folding and rolling of papers”; Rössler,
interview.
Otto E. Rössler, “Chaos in Abstract Kinetics: Two Prototypes,”
Bulletin of Mathematical Biology 39
(1977): 282.
Bruce Stewart, interview by author, New York, May 14,
2008.
Rössler, interview.
Ibid.
Ibid.
“I cannot see what thinking should be fundamentally
but just such tentative handling taking place only in the mind in some
imaginary space”; Konrad Lorenz, Der Abbau
des Menschlichen. Die Rückseite des
Spiegels. Versuch einer Naturgeschichte des Erkennens (Munich: R. Piper,
1988), 175.
See Karin D. Knorr-Cetina, Wissenskulturen. Ein Vergleich naturwissenschaftlicher Wissensformen
(Frankfurt am Main: Suhrkamp, 2002), 138.
Jacques Hadamard, An
Essay on the Psychology of Invention in the Mathematical Field (New York:
Dover, 1954), 76.
Rössler identified “all mental pictures as visual pictures
as well,” adding: “There are only visual pictures.” Rössler, interview.
Ibid.
At the end of February 1882, Nietzsche wrote to
Heinrich Käselitz (alias Peter Gast) on a typewriter that only had capital
letters: “SIE HABEN RECHT—UNSER SCHREIBZEUG ARBEITET MIT AN UNSEREN GEDANKEN”
(“You are right—our writing tools influence our thinking”; Friedrich Nietzsche,
Schreibmaschinentexte:
Vollständige Edition,
Faksimiles und kritischer Kommentar, ed. Stephan Günzel and Rüdiger
Schmidt-Grépály (Weimar: Bauhaus-Universität Weimar, Universitätsverlag, 2002),
18.
“In my room it is dead still—only the scratching of my
pen on the paper—for I love to think while writing, since the machine has not
yet been invented that registers our unexpressed, unwritten thoughts on any
kind of material”; Friedrich Nietzsche, Werke
und Briefe. Historisch-kritische Gesamtausgabe, vol. 2 of Jugendschriften 1861–1864, ed. Hans
Joachim Mette (1933–40; Munich: Deutscher Taschenbuch, 1994), 71.
Rössler, interview.
Beuys, quoted from Magdalena Holzhey, Im Labor des Zeichners: Joseph Beuys und die
Naturwissenschaft (Berlin: Reimer, 2009), 86.
Ibid., 87.
Merleau-Ponty, quoted from Georg Trogemann et al,
“Experimente,” in Code und
Material—Exkursionen ins Undingliche, ed. Georg Trogemann (Vienna: Springer,
2010), 29.
Beuys, Zeichnungen
1947–59, 17.
