snl

sdJournal of Aesthetic Education, Vol. 42, no. 4, Winter 2008 ©2008 Board of Trustees of the University of Illinois

Chaos, Fractals, and the Pedagogical Challenge of Jackson Pollock’s “All-Over” Paintings

FRAnCIs HAlsAll

Introduction

The “all-over” abstract canvases that Jackson Pollock produced between 1943 and 1951 present a pedagogical challenge in how to account for their apparently chaotic structure. One reason that they are difficult to teach about is that they have proved notoriously difficult for art historians to come to terms with. This is undoubtedly a consequence of their abstraction. In the face of an apparent disintegration of the traditional pictorial distinc-tion between figure and ground, multiplicities of (sometimes contradictory) readings present themselves. The question, posed to students, as to what they represent is an open one. To many, recalling the opinions of the baffled audience of the 1950s, they are nothing but an inchoate mess. In 1950 Time magazine referred to them as “chaos,” prompting Pollock to wire a heated reply, “nO CHAOs dAMn IT.”1

Pollock’s claim of “no chaos” can, however, be unpacked. This article looks at how a scientific analysis based on Chaos Theory and fractal pat-terns can be used to demonstrate to students that the paintings are indeed chaotic, but that this in turn provides evidence of an internal structure, an order within the chaos. This order is mimetically similar to other chaotic patterns and systems, from coastlines to economic systems. My overall posi-tion in this article is one of skepticism as to what such analysis ultimately has to offer in art historical, aesthetic, or pedagogical terms.

The Problem

For a student encountering Pollock’s work for the first time, especially after a standard first-year survey of Western art, it can seem incomprehensible.

Francis Halsall is lecturer in the history and theory of modern and contemporary art at the national College of Art and design in dublin, Ireland. He studied art history at the University of Glasgow (MA and Phd). He is the author of Systems of Art (Peter lang, 2008) and the contributing co-editor (with Julia Jansen and Tony O’Connor) of the collection Rediscovering Aesthetics (stanford University Press, 2008). He is working on a major postdoctoral project on niklas luhmann’s theory of art.

snl

sd

2 Halsall

The challenge of Pollock’s work is that none of the multiple interpretations that have been brought to bear on the paintings have any special claim to validity. Given the suggestive ambiguity and “chaos” of the images, there is little in the work itself that challenges the numerous, often contradictory, readings that have been posited. In fact, how art historians approach such images tells students something about the way art history itself works and how art historians are faced with particular choices in applying their discur-sive models. They must choose whether these accounts “fit” or satisfy the conditions of “rightness,” to evoke nelson Goodman’s terms.2

The numerous questions that have been asked of what the paintings actually represent are themselves demonstrative of the wide variety of tech-niques that modern art historians have at their disposal. Analyses have been based upon observations that are couched in the vocabularies of the politi-cal (often Marxist theory), feminism, psychoanalysis, iconography, formal-ism, and so forth.3 At the time of the paintings’ creation even the CIA were in on the act; apparently realizing the ambiguity of the work left it open to political manipulation, they secretly funded the promotion of abstract expressionism as a weapon of the cold war.4 simultaneously, members of Congress such as George dondero were actually claiming the opposite—that Pollock was making paintings that were a “means of espionage” (sic) and that “If you know how to read them . . . [they] will disclose weak spots in U.s. fortifications, and such crucial constructions as Boulder dam.”5

Ernst Gombrich’s celebrated frustration with abstract art highlights the problems that those following traditional art historical methods had with such art at the time. Writing on Pollock’s Number 12 (1952), Gombrich also seemed perplexed in the face of chaos. His response to the lack of explicit representational content or apparently accountable meaning was to claim:

It is quite consistent that these [action] painters must counteract all semblance of familiar objects or even of patterns in space. But few of them appear to realise that they can drive into the desired identifica-tion only those who know how to apply the various traditional con-sistency tests and thereby discover the absence of any meaning except the highly ambiguous meaning of traces.6

Indeed, if there is any consensus to be communicated to students about the work, it is that a single, specific meaning will always allude them. All of the above reactions to the work makes the claims of three physicists to have found the “fundamental content” of several of Pollock’s paintings as astonishing as the images themselves. In two papers that appeared in Physics World and Nature in 1999, Richard Taylor, Adam Micolich, and da-vid Jonas stated that through the application of “scientific objectivity” they could determine the specific meaning of the paintings.7 In a follow-up arti-cle that appeared in Leonardo in 2002, the claim was reiterated with the bold statement that by identifying certain fractal patterns in the paintings, the

snl

sd

Jackson Pollock’s “All-Over” Paintings 3

authors were “ending 50 years of debate over the content of his paintings.”8 To determine this fundamental content, Taylor, Micolich, and Jonas had analysed several paintings and found mathematical, fractal patterns in the chaos of paint. Thus, the “chaos” that Pollock denied was not only present but was actually evidence of a underlying structure and order that Chaos Theory could uncover.

What Is a Fractal?

stated simply, a fractal is a self-similar pattern.9 The pioneer of Fractal Geometry, Benoit Mandlebrot, described fractals as patterns that are the same at different levels of magnification: “Broadly speaking, mathematical and natural fractals are shapes whose roughness and fragmentation nei-ther tend to vanish, nor fluctuate up and down, but remain essentially un-changed as one zooms in continually and examination is refined. Hence the structure of every piece holds the key to the whole structure.”10 Importantly, a fractal need not necessarily look the same at different scales.11 despite this, it can be described using the same rules12—that is, it demonstrates statistical self-similarity,13 meaning that the same, simple mathematical rule describes the macro and micro elements of the particular complex structure. The von Koch Curve (conceived by Helge von Koch in 1904) is one of the simplest ways of imagining a fractal (see figure 1). It is a pattern that follows the simple rule of repetition of a simple triangle shape.14 Its construction is described by Richard Voss thus:

Figure 1. The von Koch curve represented at different stages of its genesis.

snl

sd

4 Halsall

A simple line segment is divided into thirds and the middle segment is replaced by two equal segments forming part of an equilateral tri-angle. At the next stage in the construction each of these four seg-ments is replaced by four new segments with length 1/3 of their par-ent according to the original pattern. This procedure, repeated over and over, yields the beautiful von Koch curve.15

Therefore, by the continued iteration of the initial rule of construction, an image emerges that will be infinitely complex in so far as it will obey this simple rule regardless of the scale of magnification at which it is viewed. The edge of the von Koch curve, comprised as it is of ever diminishing tri-angles, is a loop that tightens in upon itself to a greater degree the higher the magnification. Further, just as the detail of the edge of the image will con-tinue to increase as the scale of magnification of the image increases to infin-ity, so too the length of the line around the space enclosed by the edge of the fractal is infinite. Therefore, if we take the initial triangle from which the curve is extrapolated as having sides of length 1, then, as Gleick explained, “The length of the boundary is 3 x 4/3 x 4/3 x 4/3 . . . .—infinity. Yet the area remains less than the area of a circle drawn around the original triangle.”16 This highlights a key paradox of the fractal: the incongruity between the area of the fractal, which can be shown to be less than the area of a circle drawn around the whole image, and the possibility that this finite area is contained within an infinitely long edge. Fractals are most likely to appear in complex and dynamical systems that are poised in a state of nonequilibrium—that is, within systems that are de-scribed by Broomhead and Jones as ones whose “dynamical behaviour is governed by non-linear evolution equations.”17 This includes such dynami-cally chaotic systems such as those that have been observed in the natural world, including those seen in connection with neuron activity (in conscious systems), chemical reactions, fluid flows, and economic systems, all of which have been modelled using the methods of fractal geometry.18 Further examples of this are provided by Gleick, who related how problems in the mapping of human anatomy have been solved through the application of fractals. A few examples serve to illustrate the wide variety of systems that display fractal patterns; the urinary collecting system is constructed accord-ing to a fractal structure,19 as are the edges of clouds and the economic sys-tem of twentieth-century cotton prices as analyzed by Mandelbrot.”20

In short, despite the visual complexity of fractal patterns, they conform to the simple rules of their construction regardless of the scale of magnifica-tion at which they are viewed. In other words, the pattern is underpinned by a structure that can be explained by a few, relatively simple, rules. The implication of this is that if a similar type of pattern could be discovered in Pollock’s paintings, then it could be demonstrated that his protesta-tions of “nO CHAOs dAMn IT,” could be verified by proving that there

snl

sd

Jackson Pollock’s “All-Over” Paintings 5

was an underlying structure to the images after all—and one that could be explained in simple terms.

Pollock and Fractals

To demonstrate the existence of fractals within Pollock’s paintings, Jonas et al. analyzed several paintings. The chosen paintings were scanned into a computer, overlaid with a computer-generated grid of identical squares, and analyzed at various levels of magnification from squares of length 0.8 millimeters (the detail of the very finest paintwork) to squares of around 2.5 meters (a square containing the whole canvas).21

Jonas et al. found that in several of the paintings they investigated there were two co-existent fractal patterns: the first pattern corresponding to the fine paintwork at a scale between 1mm and 5cm; and the second pattern corresponding to the broad paintwork at a scale between 5cm and 2.5 m.22 Both patterns can be expressed mathematically as what is called the “frac-tal dimension” of the pattern. Fractal dimensionality is obscure. It is the method Mandlebrot used to describe the relationship the viewer has to the object with regard to the scaling, or the level of detail at which the object is analyzed. It expresses the level of detail at which the object is self-similar in relation to the observer’s level of scrutiny.23 Contrary to the three dimen-sions of Euclidean geometry—height, depth, and breadth—of which we are intuitively familiar, fractal dimensions need not be an integer. Fractal dimension is a way of expressing a level of detail of a self-similar shape in relation to the observer’s level of scrutiny. Thus, the fractal dimension, or d, provides an expression of the level of detail of that shape, or, as Voss has explained, “the fractal dimension d, thus, provides a quantitative measure of the wiggliness of the curves.”24 For certain shapes, Mandlebrot argued, their dimensionality is best expressed in terms between integers; they are, in other words, not clearly defined by one scale but instead have a certain, yet characteristic, irregularity that remains constant over different scales. This is the regular irregularity of the self-similar and infinite patterns suggested by fractals.25 In short, the fractal dimension, or d, provides an expression of the level of detail of that shape. Thus, for example, Blue Poles Number 11 (1952) was found to demonstrate two fractal patterns, the first of which the close detail fractal, or dd, having a value 1.72, with the larger scale pattern, or dl, having a value that was stat-ed to be “quite close to 2.” In all the paintings analyzed, the second, large-scale fractal was found to have a value close to 2, and the following values were found for the small-scale fractal pattern, dd: Untitled: Composition With Pouring II (1943) has a dd value “close to 1”; Number 14 (1948) has a dd value of 1.45; and Autumn Rhythm: Number 30 (1950) has a dd value of 1.67.26

snl

sd

6 Halsall

The question, at this point, should be, But what does this prove? It is worth recalling that Jonas et al. had claimed that through the application of “scientific objectivity” the holy grail of art historical investigation could be uncovered—namely. the “fundamental content” of the images. In other words, the fundamental content of Autumn Rhythm (Number 30) is the two fractal dimensions:

1. a close detail fractal, or dd of value 1.67

2. a larger scale fractal pattern, or dl of a value that was stated to be “quite close to 2”

This is reminiscent of the statement made by the computer deep Thought in douglas Adams’s Hitch-Hikers Guide to the Galaxy, which having pon-dered the ultimate existential question, the meaning of life, the universe, and everything, gave the answer as being “forty-two.” Upon giving the underwhelming answer, the computer subsequently made the Earth, as a computer, to calculate the Question of what was the meaning of life, the uni-verse, and everything. And like deep Thought, it would seem that if the fundamental content of Autumn Rhythm (Number 30) is 1.67 and nearly 2, then the questions we are asking of it need some further investigation and refinement. This question of what is being represented by such fractal pat-terns needs further unpacking. In the remainder of this article I will outline four different accounts that might be presented to students as to what the existence of fractal patterns within Pollock’s paintings demonstrate by asking the following question: What can be observed in the paintings?27

What Can Be Observed? (1) Nature/Mimesis/Representation

First, it could be argued that the existence of fractal patterns in Pollock’s all-over canvases provide evidence of structures in the work that are mimet-ic of naturally occurring forms. This is because they correspond to naturally occurring fractal patterns. As has already been observed, fractal patterns oc-cur naturally in nature (for example, in bracken fronds and capillaries in the lungs). Jonas et al. note that natural fractal patterns such as coastlines and lightning display a fractal dimension of between 1.25 and 3. They also observe that systems that have two fractal dimensions, like the paintings, are also naturally occurring. The examples they provide of this are “trees and bronchial systems.” Thus, by illustrating fractal patterns and demon-strating that such dynamic, systemic patterns are naturally occurring, the paintings could be seen to be mimetically representative of naturally occur-ring dynamic systems. Hence, the dynamism of the natural world is seen as re-presented within the complex artistic systems of the paintings. This

snl

sd

Jackson Pollock’s “All-Over” Paintings 7

certainly supports a popular reading of Pollock’s work—that it is somehow natural—which Pollock himself evoked with titles such as Autumn Rhythm. For example, Martin Gardner has suggested that “photographers with a keen sense of aesthetics find it easy to take photomicrographs of natural patterns that are almost indistinguishable from abstract expressionist art.”28 Other art historical readings have suggested that the patterns of seaweed Pollock saw while mussel farming on the beach on long Island have worked their way into work such as Blue Poles.29

What Can Be Observed? (2) The Fractal Itself

The second interpretation is a more rigidly formalist reading of the fractal patterns. It suggests that what is being represented is the fractal itself. This question takes the painting as a closed system of representation gesturing toward the conceptual object of a mathematical model, the icon of which is the fractal pattern. This second approach brings the response within the sphere of the claims that have been made for other fractal images. If Pollock’s paintings are frac-tals, then naturally it follows that what has been claimed for fractal images may be claimed for the paintings as well. Fractal images provided as repre-sentations of mathematical models have been applied to an aesthetic analy-sis, as if they were artworks. And it has been suggested that in their graphi-cal representation of the nonrepresentable infinite regress of self-similarity, they might display aesthetic qualities. Certainly this is how Mandelbrot conceived of his fractal geometry, as a “world of pure plastic beauty unsus-pected till now.”30 In this respect, then, it could be argued that fractals pro-vide an ultra-modernist art form: an expression of Mondrian’s Pure Plastic Art. Mondrian’s definition of his “Pure Plastic Art” would indeed seem to fulfill the criteria of a fractal insofar as, he argued, that “The important task of all art, then, is to destroy the static equilibrium by establishing a dynamic one. non-figurative art demands an attempt of what is a consequence of this task, the destruction of particular form and the construction of a rhythm of mutual relations, of mutual forms of free lines.”31

Following this line of argument, the self-similar, fractal patterns of Autumn Rhythm (Number 30), as with other visual representation of fractals (such as the image of a Julia set), suggest a visual system that is in a state of dynamic, nonequilibrium, and that, by virtue of its scaling self-similarity, is a “construction of a rhythm of mutual relations, of mutual forms of free lines.” Couched in these terms, fractals (Pollock’s included) are exemplary of modernist Pure Plastic Art. developing this aesthetic analysis further, in an article that appeared in Modern Painters P. W. Atkins argued that images based upon fractals ful-filled the description of being a “hypermodern art.” He argued that in its

snl

sd

8 Halsall

synthesis of aesthetic and scientific concerns the fractal could provide the conceptual basis for an art that was suitable for our technologically and scientifically advanced society. From his starting point that “it is plausible that a rigorously scientific basis of aesthetics will one day be available,” he argued that aesthetic response results from the “exposure” of the “appro-priately resonant circuitry”32 of the brain to the aesthetic object that is the focus of the aesthetic attention. Atkins argues that such a model of aesthetic response, which is reminiscent of the “free-play” of reason and imagina-tion in Kantian aesthetics, can be conceived of in materialist terms as be-ing a function of the organic system of consciousness (although Atkins also admits that such modelling is not yet possible). In conclusion, Atkins goes on to state that because fractals are mathematically, intellectually, and aes-thetically both engaging and pleasing, they are “beautiful” images. This is because, he argued, “At one level the warmth of our response to it must lie in the dizzying richness of the image. A fractal image is an image without end.” Fractals are, for Atkins, in their simple, yet theoretical infinite com-plexity, “deep, inexhaustible wells of beauty.”33

Following this line of thought it could be argued that as fractals Pollock’s paintings such as Autumn Rhythm (Number 30) also fulfill the strictly for-mal (and mathematical) conditions of being “deep, inexhaustible wells of beauty” and are, therefore, a fulfillment of the criteria of a hypermodern art as described by Atkins. The logical conclusion of this would seem to sug-gest that if fractals are inherently aesthetic, and the canvases are inherently fractal, then it is possible to provide an algorithm for beauty in art. At the heart of this take on fractals is the suggestion that there can be a mathemati-cally formalized set of formal procedures (algorithms) lying at the core of the visual systems of Pure Plastic, hypermodern art. This plays like both a parody of Greenberg’s formalism, in which the beauty of Pollock’s work is explained by the inherent form that the underlying structure suggests, and an inversion of Kant’s argument that mathematical forms cannot be beauti-ful because of their self-similarity.

What Can Be Observed? (3) An Index of Process

The third interpretation of the fractal patterns in the paintings relates to their patterns as an index of the process of their creation. As Rosenberg suggested in “The American Action Painter’s,” one way of looking at the complex vi-sual system of Pollock’s paintings is that what is being represented on the canvas is “not a picture but an event.”34 Thus, the patterns are representa-tive of Pollock’s movements around the canvas and hence serve as a visual record of not only his own movements around the canvas but also the ap-plication of the paint itself, whether it was placed by Pollock or allowed to fall by its own devices. That this index of creation and the application of the

snl

sd

Jackson Pollock’s “All-Over” Paintings 9

paint is fractal suggests that the way in which the paint fell was chaotic and hence dynamic in its nature, in the same way that other natural process can be modelled using the visual analogue of the fractal.35

As already noted in the paintings analyzed by Jonas et al., two fractal dimensions were identified, corresponding to two different types of pat-terns suggesting two processes of construction. Jonas et al. acknowledge this and suggest that the two fractal dimensions represent the following:

(i) An all-over, large-scale pattern that could be seen as the visual scaffolding of the paintings. This corresponds to Pollock’s movement around the canvas. This pattern was identified as having a fractal dimension of “nearly 2.” significantly, this figure is the same for the larger pattern in all of the canvases that were analyzed. This suggests that a similar process was being employed each time. This pattern is the anchor layer of the painting, upon which the complete pattern is built upon, around, and within.

(ii) A smaller-scale fractal pattern that fills in the detail of the scaffolding of the anchor layer. This pattern is an index of a finer, more detailed working approach that concentrates upon particular areas of the can-vas at any one time (as opposed to the anchor layer, which involves working on the canvas as a whole). It is suggested that this pattern is the detail layer of the painting. Taylor, Jonas, and Micolich claim that this smaller-scale pattern represents Pollock’s dripping tech-nique, which was “refined” between the years 1943 and 1952: “Our analysis shows that Pollock refined his dripping technique: the frac-tal dimensions increased steadily through the years from close to 1 in 1943 to 1.72 in 1952.”36 They argued that this increase in the fractal dimension, which means simply that the pattern became more expan-sive, meant that the pattern of this detail layer became more dense in Pollock’s later abstract work. This could be used, they argue, as a legitimate art historical technique: “fractal analysis could be used as a quantitative, objective technique both to validate and date Pollock’s drip paintings.”37

The claim that these two fractal patterns inherent in Pollock’s visual systems correspond to different working practices can be validated by look-ing at accounts of Pollock’s method of working on the abstract canvases. These accounts corroborate the argument that the patterns are visual indica-tors of the dynamic events of their creation. Firsthand accounts of Pollock at work by Goodnought, namuth (who filmed Pollock), and Karmel38 (who has produced an exhaustive study based upon namuth’s films and other accounts of Pollock at work) demonstrate, on one the hand, Pollock’s broad approach to the whole canvas (corresponding to the fractal of the anchor

snl

sd

10 Halsall

layers) and, on the other hand, his more detailed close working of the canvas ( corresponding to the fractal of the detail layers). The accounts of the processes involved in the creation of Autumn Rhythm (Number 30) demonstrate this. In an article that first appeared in Art News in 1951, Goodnought provided a detailed description of Pollock’s method. At the start of the process Pollock began with a blank, clean canvas, which, Goodnought observed, was worked in a period of intense and absorbing activity (about two hours) at the end of which Pollock was utterly exhausted. Goodnought continues:

After a while he took a can of black enamel . . . and a stubby brush which he dipped into the paint and then began to move his arm rhythmically about, letting the paint fall in a variety of movements on the surface. At times he would crouch, holding the brush close to the canvas, and again he would stand and move around it or step on it to reach to the middle. Within half an hour the entire surface had taken on an activity of weaving rhythms . . . As he continued, still with black, going back over former areas, rhythms were intensified with counteracting movements.39

Then, Goodnought related, Pollock would spend two weeks becoming accustomed to the painting, after which he would sporadically apply differ-ent layers of paint to the canvas in a more deliberate fashion. The third and final working of the canvas, some weeks after the initial act of construction, is described by Goodnought in terms that are instructive. This process was described as being “slow and deliberate. The design had become exceedingly complex and had to be brought to a state of complete organization.”40

While Karmel discusses inconsistencies with Goodnought’s account, it is clear nonetheless that Pollock did work at different levels of intensity at different periods of time until the canvas as a whole was left. Goodnought’s identification of three types of activity (the initial work, the complementary changes, and the final, close detailing) would, therefore, correspond with Taylor, Jonas, and Micolich’s argument that the different fractal dimensions represent different working processes—on the one hand a canvas-wide ac-tion, and on the other a closer detailing of the pattern. Karmel also provides evidence of this based on his close scrutiny of the two films namuth made of Pollock at work. To demonstrate this Karmel uses some of namuth’s col-or film outtakes from the filming of Pollock working on a now lost painting made on red canvas. Karmel suggests that this footage refutes the idea that the paint is applied rapidly at random; instead, he says, a “step by step evo-lution of the painting from bare canvas to a complex web” takes place, which demonstrates Pollock’s nonrandom “combination of kinetic freedom and formal control.”41 As Karmel also observes with reference to one sequence of film:

snl

sd

Jackson Pollock’s “All-Over” Paintings 11

A close look at several frames [of the film] sequence underscores the fact that the splats are produced by a different hand motion than the ini-tial curves. After taking the brush from the paint can, Pollock lifts his hand and bends the brush inward, toward his body; he then snaps his forearm downward, finishing with his palm down rather than up. It is the same gesture of wind-up and release visible in some of the still photographs of Pollock working on Autumn Rhythm.42

such a working method is also seen in the section of the second film where Pollock paints onto glass, with the camera positioned beneath the glass. There are two important parts of this statement. First, the observation that there is visual evidence of Pollock’s differing techniques again corrobo-rates the argument that the two fractal patterns correspond to two differing types of activity. second, there is the observation that Pollock used the same type of motions between canvases (in this case the lost, red canvas painting and Autumn Rhythm (Number 30), which was subjected to fractal analysis). This further confirms the argument that there is a fractal similarity shared between (a selection of) the canvases.

Conclusion

Thus far I have identified three scales of observation that can be applied to the interpretation of the fractal patterns in Pollock’s paintings. These are natural, Aesthetic, and Process. My conclusion is to offer a skeptical response to the art historical, aesthetic, and, most importantly, pedagogi-cal effectiveness of the above analyses. All three of the these interpretations are, at heart, formalist. They deal with the specific formal qualities of the paintings, and the formal analysis reveals fractal patterns. However, a more satisfactory reading of Pollock’s work will move to a different scale of obser-vation. An effective account of any work of art must extend its focus beyond the singular work itself to the historical and political contexts of its produc-tion and reception and the social systems within which it is situated. Fur-ther, the problem with the fractal analysis becomes a conceptual one when it is claimed, as Jonas et al. do, that this is the “fundamental content” of the images. There are two problems here:

1. A problem of inversion: The scientists are using fractals to represent Pollock’s paintings while claiming that the paintings represent frac-tals. We know specifically that Pollock did not attempt to represent fractals. Instead, the fractal patterns have been used as an interpretive framework by which re-present the paintings, just as you can use frac-tal patterns to describe and re-present coastlines and bracken fronds. Yet coastlines and bracken fronds do not represent anything.

snl

sd

12 Halsall

2. A problem of appropriateness: To treat a Pollock’s paintings as if they were a coastline is to merely deliver a interpretation of their math-ematical structure without actually saying anything particularly interesting about them. such a strategy leads, ultimately, to histori-cally sterile claims about the paintings based on their numerical val-ues. In short, fractal analysis as a way of worldmaking does not “fit” the paintings.

I began by claiming that the difficultly as well as the radicalism of Pollock’s work lies in its inchoate and formless chaos. His importance in the development of the avant-garde lies in his dismantling of the traditional basis of pictorial representation, namely, the distinction between figure and ground. As Clement Greenberg argued in “The Crisis of the Easel Picture” (1948), Pollock’s importance as a painter is that he suggested “a way be-yond the easel, beyond the mobile, framed picture.”43 Greenberg’s reading of Pollock’s canvases as flat planes, or all-over pictorial fields, means that the perceptual difference between figure and ground is negated. There are two dominant readings of this. First, in Greenbergian terms, this places Pollock at the end of a developmental narrative of modernist painting that ends with the move from representation to the articulation of the fundamental flatness of the medium of painting. However, this was a point beyond which modernist painting historically did not develop. second, the effacement of figure by ground also places Pollock within the narrative of the informe, formless, or antiform where, as Krauss has argued, Pollock’s dismantling of the conventions of the easel picture was subsequently investigated in further media by Robert Morris, Richard serra, and Andy Warhol (amongst others.) The fractal analysis, however, re-inscribes traditional pictorial and aesthetic values within Pollock’s paint-ings. It does so by re-inscribing resemblance back into the images by read-ing the fractal as an identifiable figure against the ground of the pictorial spaces behind it. The re-emergence of the figure/ground distinction thus re-inscribes Pollock within the tradition of representational easel painting rather than investigating the historical conditions by which he heralded its dissolution.44 And herein lies a certain irony—namely, the point at which Pollock’s paintings become formless and chaotic is when they become avail-able to fractal analysis. Or, in other words, when figure and ground are ef-faced in one representative order, they re-appear in another. This paradox is illustrated further by a particular example from Karmel. Karmel finds an initial figurative order to Pollock’s visual system. In con-trast to Goodnought’s account of Pollock’s tripartite working method, Karmel suggests the work has four stages, with the first, intense, period of activity being split into two distinct actions. First there was an initial trac-ing of calligraphic and loosely representational figures (clearly distinct from the ground), which was then followed by the second task of immediate

snl

sd

Jackson Pollock’s “All-Over” Paintings 13

over-painting and obscuring of these figures. In one example, Number 27, 1950, the painting was begun with black lines and a series of loose arcs, which are vaguely descriptive of the Picasso-esque motifs that appear else-where in Pollock’s work. Karmel has related this inception of the work:

Watching from above, we see a hand extend a paint-clotted brush a few inches above the bare canvas. A black line appears on the white surface below it. As the brush moves through the air, the line swerves to the left, curves upward, and then loops to the right, forming a pumpkin shaped ovoid. narrow, then thick, then narrow again, it ter-minates in a pair of tiny blots connected by a stem; a miniature bar-bell with one end light and the other dark. . . .

In the following nine seconds, Pollock adds a series of lines describing the contours of a figure unmistakably similar to the figures in earlier pictures such as Untitled (Cut Out) (1948). One line curves upward on the left to form a shoulder then turns down and right to make an arm, terminating in calligraphic fingers. A second line rises from the left corner of the canvas to describe a foot and a leg. A third descends on the right, bending at the knee. A fourth is bent into a foot, shaped like a triangular shirt hanger. Moving right, Pollock begins a second figure, working upward this time instead of down.45

Having stated this Karmel argues next that the randomness of Pollock’s painting in the second half of this first stage obscures this initial represen-tational aspect of the picture. This second stage, he continued, “is the point at which the painting is transformed from a collection of independent pic-tographs into a single all-over composition united by a consistent rhythm of dark and light, thick and thin extending across the surface.”46 Thus, the point at which the question—What is being represented?—becomes impossi-ble to answer in one way, another mode of answering it emerges This is be-cause the point of abstraction is the point at which the fractal figure emerges from the ground. The irony is that by being nonrepresentative, the paintings end up becoming representative again on a different scale. I think we should encourage students to hold onto the meaningless and chaos of such pictures. Arguably, the power of Pollock’s work, and hence his continued importance in the development of avant-garde art, is a very func-tion of its abstraction. That is, the paintings will resist comprehension by re-sisting representation. This means that attempts to find singular definitions of the representative content of the paintings will always be confounded by the works’ complexity.

nOTEs

some parts of this paper were presented at the conference Beyond Mimesis and nominalism: Representation in Art and science, June 22-23, 2006, lsE/ Courtauld, london.

snl

sd

14 Halsall

1. “In 1950, Time magazine printed parts of an essay by Bruno Alfieri, previously published in Italy, which used the word ‘Chaos’ to describe the paintings of Jackson Pollock. Unable to let the criticism pass, Pollock sent Time a telegram that began, ‘nO CHAOs dAMn IT.’” From James Coddington, “no Chaos damn It,” in Jackson Pollock: New Approaches, ed. Pepe Karmel and Kirk Varnedoe (new York: Museum of Modern Art, 2000), 101. Coddington discussed the or-der of Pollock’s paintings in relation to his method but does not reference chaos theory, fractal analysis, or systems theory in doing so.

2. nelson Goodman, Ways of Worldmaking (Indianapolis: Hackett, 1978), 17, 19. 3. For a selection of the recent reception of Pollock see Karmel and Varnedoe,

Jackson Pollock: New Approaches; and M. leja, Reframing Abstract Expressionism: Subjectivity and Painting in the 1940s (new Haven, CT: Yale University Press, 1993).

4. see Frances stonor saunders, Who Paid the Piper? The CIA and the Cultural Cold War (london: Granta, 2000); E. Cockfroft, “Abstract Expressionism, Weapon of the Cold War,” Artforum (June 1974); s. Guilbaut, How New York Stole the Idea of Modern Art (Chicago: University of Chicago Press, 1983).

5. “dondero’s neurotic assessment was echoed by a coterie of public figures, whose shrill denunciations rang across the floor of Congress and in the conservative press. Their attacks culminated in such claims as “ultramodern artists are un-consciously used as tools of the Kremlin” and the assertion that, in some cases, abstract paintings were actually secret maps pinpointing strategic U.s. fortifica-tions. ‘Modern art is actually a means of espionage,’ one opponent charged. ‘If you know how to read them, modern paintings will disclose the weak spots in U.s. fortifications, and such crucial constructions as Boulder dam’”( saunders, Who Paid the Piper, 253-54).

6. E. Gombrich, Art and Illusion (london: Phaidon, 1960), 244. 7. Richard Taylor, Adam Micolich, and david Jonas, “Fractal Expressionism,”

Physics World, October 10, 1999; Richard Taylor, Adam Micolich, and david Jonas, “Fractal Analysis of Pollock’s drip Paintings,” Nature, June 3, 1999, 422.

8. Richard Taylor, Adam Micolich, and david Jonas, “The Construction of Jackson Pollock’s Fractal drip Paintings,” Leonardo 35, no. 2 (2002): 203-7.

9. The literature on fractals is vast. The best introduction to the often baffling the-ory and application of fractal geometry is found in James Gleick, Chaos (new York: Vintage, 1998). The classic work on fractals is Benoit Mandlebrot, The Frac-tal Geometry of Nature (new York: W. H. Freeman and Company, 1977), which is a balance between accessible descriptions of the implications of the theory and obscure mathematics. More recent discussions on the applied use of fractal geometry include M. Fleischmann, d. J. Tildesley, R. C. Ball, eds., Fractals in the Natural Sciences (Princeton, nJ: Princeton University Press, 1989); Heinz-Otto Peitgen and dietmar saupe, eds., The Science of Fractal Images (new York: spring-er Verlag, 1988); Heinz-Otto Peitgen, The Beauty of Fractals (new York: springer Verlag, 1986).

10. B. B. Mandlebrot, “Fractal Geometry: What Is It, and What does It do?” in Fractals in the Natural Sciences, ed. M. Fleischmann, d. J. Tildesley, and R. C. Ball (Princeton, nJ: Princeton University Press, 1989), 3-16.

11. In the terms of chaos theory, this self-similarity is expressed in terms of the shape fulfilling the same mathematical description rather than being visually identical.

12. “The key to fractal geometry’s effectiveness resides in a surprising discovery that the author has made, largely thanks to computer graphics. The algorithms that generate the other fractals are typically so extraordinarily short, as to look positively dumb. This means that they must be called ‘simple.’ Their fractal outputs, to the contrary, often appear to involve structures of great richness” (Mandlebrot, “Fractal Geometry,” 6).

13. Richard Voss, “Fractals in nature: From Characterization to simulation,” in The Science of Fractal Images, ed. Heinz-Otto Peitgen and dietmar saupe (new York: springer Verlag, 1988), 30.

snl

sd

Jackson Pollock’s “All-Over” Paintings 15

14. The von Koch curve is so named after the Helge von Koch who first described the pattern in 1904.

15. Voss, “Fractals in nature,” 26.16. Gleick, Chaos, 99.17. d. s. Broomhead and R. Jones, “Time-series Analysis,” in Fractals in the Natural

Sciences, ed. M. Fleischmann, d. J. Tildesley, and R. C. Ball (Princeton, nJ: Princ-eton University Press, 1989), 103.

18. see especially P. Fischer and William R. smith, eds., Chaos, Fractals and Dynamics (new York: Marcel dekker, 1985) and Peitgen and saupe, The Science of Fractal Images.

19. Gleick, Chaos, 109. Other examples can be found in Peitgen and saupe, The Sci-ence of Fractal Images; and Fleischmann, Tildesley, and Ball, Fractals in the Natural Sciences.

20. “Indeed when Mandlebrot sifted [Houthkakker’s] cotton-price data through IBM’s computers, he found the astonishing results he was seeking. The numbers that produced aberrations from the point of view of normal distribution [and the classical economical model of supply and demand] produced symmetry from the point of view of scaling. Each particular price change was random and un-predictable. But the sequence of changes was independent of scale; curves for daily prices and monthly price changes matched perfectly. Incredibly, analyzed Mandelbrot’s way [according to fractal patterns] the degree of variation re-mained constant over a tumultuous sixty-year period that saw two World-Wars and a depression” (Gleick, Chaos, 86).

21. Taylor, Micolich, and Jonas, “Fractal Expressionism,” 25-28.22. “This fractal analysis reveals two distinct d values occurring over the ranges

1mm l < 5cm and 5 cm < l 2.5 M” (Taylor, Micolich, and Jonas, “Fractal Analysis of Pollock’s drip Paintings,” 422).

23. Peitgen and saupe, The Science of Fractal Images, 29. Gleick describes this concept of fractal dimensions thus:

What is the dimension of a ball of twine? Mandlebrot answered, it de-pends on your point of view. From a great distance the ball is no more than a point, with zero dimensions. From closer the ball is seen to fill spherical space, taking up three dimensions. From closer still, the twine comes into view, and the object becomes effectively one-dimensional, though the one dimension is certainly tangled around itself in a way that makes use of three dimensional space. The notion of how many numbers it takes to specify a point remains useful. From far away it takes none—the point is all there is. From closer it takes three. From closer still, one is enough—any given position along the length of twine is unique, whether the twine is stretched out or tangled up in a ball. (Gleick, Chaos, 95)

24. Peitgen and saupe, The Science of Fractal Images, 29.25. “The large amount of repeating structure within a fractal pattern causes it to

occupy more space than a smooth on-dimensional line, but not to the extent of completely filling the two dimensional plane” (Taylor, Micolich, and Jonas, “Fractal Expressionism,” 25-28).

26. Ibid., 25-28.27. Because each of these four suggestions engage in a different scale of observation,

they also function as an analogy for the form of the fractal itself—that is, as a self-similar pattern that is manifested at different scales within the discursive system of this paper.

28. larry short, “The Aesthetic Value of Fractal Images,” British Journal of Aesthetics 31, no. 4 (October 1991): 351.

29. steven naifeh and Gregory White smith, Jackson Pollock: An American Saga (new York: Clarkson n. Potter, 1989).

30. Mandelbrot, The Fractal Geometry of Nature, 4.31. Piet Mondrian, “Pure Art and Pure Plastic Art” (1936), in Theories of Modern Art,

ed. Herschel B. Chipp (Berkeley: University of California Press, 1968), 349ff.

snl

sd

16 Halsall

32. P. W. Atkins, “The Rose, the lion, and the Ultimate Oyster,” Modern Painters 2, no. 4 (Winter, 1989/90): 50-55.

33. A similar line of argument is pursued in short, “The Aesthetic Value of Fractal Images,” 342ff.

34. Harold Rosenberg, “American Action Painters,” in Art in Theory 1900-1990, ed. Charles Harrison and Paul Wood (london: Blackwell, 1991).

35. Whirlpools and sand piles are two such examples discussed by Mark Buchanan in Ubiquity (london: Wiedenfeld and nicolson, 2000).

36. Taylor, Micolich, and Jonas, “Fractal Analysis of Pollock’s drip Paintings,” 422.37. Ibid.38. All quoted in Pepe Karmel, “Pollock at Work: The Films and Photographs of

Hans namuth,” in Jackson Pollock, ed. Kirk Varnedoe with Pepe Karmel (new York: Henry n. Abrams, 1998), 94-95.

39. Ibid.40. Goodnought quoted Varnedoe and Karmel, Jackson Pollock, 95 (emphasis

added).41. Ibid.42. Ibid., 114.43. Clement Greenberg, “The Crisis of the Easel Picture,” Partisan Review (April

1948). Reprinted in John O’Brien, ed., Clement Greenberg: The Collected Essays and Criticism (Chicago: University of Chicago Press, 1986), 224.

44. “If I am going over this all-too-familiar-ground, it is to underscore the stakes involved in promoting the idea of Pollock as a draftsman, of deciding to read his line as contour rather than its dissolution” (R. Krauss, “The Crisis of the Easel Picture,” in Jackson Pollock: New Approaches, ed. Pepe Karmel and Kirk Varnedoe [new York: Museum of Modern Art, 1999], 155-79).

45. Karmel, “Pollock at Work: The Films and Photographs of Hans namuth,” 107.46. Ibid.