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Pattern Recognition Letters 28 (2007) 695–702

Authenticating Pollock paintings using fractal geometry

R.P. Taylor a,*, R. Guzman a, T.P. Martin a, G.D.R. Hall a, A.P. Micolich b, D. Jonas b,B.C. Scannell a, M.S. Fairbanks a, C.A. Marlow a

a Physics Department, University of Oregon, Eugene, OR 97403, USAb School of Physics, University of New South Wales, Sydney, NSW 2052, Australia

Available online 13 October 2006

Abstract

Jackson Pollock’s paintings are currently valued up to US$75 M, triggering discussions that attributation procedures featuring sub-jective visual assessments should be complimented by quantitative scientific procedures. We present a fractal analysis of Pollock’s pat-terns and discuss its potential for authenticity research.� 2006 Published by Elsevier B.V.

Keywords: Abstract art; Jackson Pollock; Authenticity; Fractals

1. Introduction

In March 1952, Jackson Pollock (1912–1956) laid downthe foundations of his masterpiece Blue Poles: Number 11,1952 by rolling a large canvas across his studio floor anddripping fluid paint from an old can with a wooden stick(Varnedoe, 1998). The painting, shown in Fig. 1, representsthe culmination of 10 years of development of his ‘‘pour-ing’’ technique. In contrast to the broken lines painted byconventional brush contact with the canvas surface, Pol-lock poured a constant stream of paint onto his horizontalcanvases to produce continuous trajectories. Twenty yearslater, when the painting sold for US$2 M, only works byRembrandt, Velazquez and da Vinci commanded higherprices. Pollock’s works continue to grab attention, as wit-nessed by the success of the retrospectives during 1998–1999 when prices of US$40 M were discussed for Blue

Poles: Number 11, 1952. As the commercial worth of Pol-lock’s paintings continue to soar, judgments of authenticityhave become increasingly critical. If a poured painting ofunknown origin is found, such as the one shown inFig. 2, what methods should be employed to decide if it

0167-8655/$ - see front matter � 2006 Published by Elsevier B.V.

doi:10.1016/j.patrec.2006.08.012

* Corresponding author. Tel.: +1 541 346 4741; fax: +1 541 346 5861.E-mail address: rpt@darkwing.uoregon.edu (R.P. Taylor).

is a long-lost masterpiece or a fake? When dealing withsuch staggering commercial considerations, subjectivejudgments attempting to identify what art scholars callthe ‘‘hand’’ of the artist – the tell tail visual trademarks –may no longer be fully adequate. In particular, subjectivejudgments have become increasingly difficult to defendagainst potential litigation (Spencer, 2004). What is becom-ing clear is that subjective assessments should be coupledwith more quantitative and objective scientific investi-gations.

Unification of artistic and scientific investigations is nota new proposal for authenticity research. Investigations ofpaintings of unknown origin often call on a diverse rangeof consultants. From the arts, provenance studies (wherean art historian judges the painting’s history relative toknown facts about the artist) are coupled with connoisseur-ship (where an art expert compares a visual inspection ofthe painting with the catalog of known paintings). Fromthe sciences, a range of techniques can be employed to dateor determine the material composition of the paint, canvasand frame (Coddington, 1999; McCrone, 2001; Spencer,2004). For many artists, this combination of research toolsyields compelling evidence for attributing paintings. Unfor-tunately, Pollock’s unique history adds uncertainty to thesestudies. For example, financial success arrived late in hislife, forcing him to barter paintings (Naifeh and Smith,

Fig. 2. A poured painting of unknown origin (enamel on canvas,64.0 cm · 101.6 cm) submitted by a private collector for fractal analysis.

Fig. 1. Blue Poles: Number 11, 1952 (enamel and aluminum paint oncanvas, 210.4 cm · 486.8 cm) was painted by Pollock in 1952 (TheNational Gallery of Australia, Canberra, Australia). � 2006 ThePollock-Krasner Foundation/ Artists Rights Society (ARS), New York.Reproduction, including downloading of Pollock works is prohibited bycopyright laws and international conventions without the express permis-sion of the Artists Rights Society, New York.

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1989; Potter, 1985). These casual, unrecorded exchanges ofpaintings create a challenge for provenance judgments –long-lost Pollock paintings may exist where a clear histor-ical link between the artist and the current owner is beyondtrace. Furthermore, Pollock was the subject of unprece-dented publicity at his peak, including a film documentary(Namuth, 1980) that exposed his pouring technique to thepublic. The resulting wave of Pollock imitators is well doc-umented and provides proof of the existence of non-Pol-lock poured paintings dating from Pollock’s era andcomposed using similar paints and canvases. This limits sci-entific techniques that aim to distinguish paintings basedon material composition and age. These complicationsemphasize the importance of scientific pattern analysisfor authenticity research.

In this Letter, we build on our earlier discovery that hispoured trajectories are composed of fractal patterns (Tay-lor et al., 1999a,b). Whereas our initial studies featured acomputer analysis of five paintings, subsequent investiga-tions by ourselves and others (Mureika et al., 2005; Mure-ika, 2005) have examined 50 Pollock paintings to date, allof which have been shown to be fractal. We previouslydescribed his style as ‘‘Fractal Expressionism’’ to distin-guish it from computer-generated fractal art (Taylor

et al., 1999b). Fractal Expressionism indicates an abilityto generate and manipulate fractal patterns directly. Thediscovery raised a critical question that triggered consider-able debate: how did Pollock manage to paint such intri-cate fractal patterns, so precisely, and do so 25 yearshead of their scientific discovery in Nature? Some art schol-ars interpreted these achievements in terms of remarkableartistic talent, while others proposed that fractals are per-haps an inevitable consequence of pouring paint. Here wepresent evidence showing that fractals arise from the spe-cific pouring technique developed by Pollock. We considerwhether this ability is unique to Pollock and explore thepotential of fractal analysis techniques as an authenticitytool. First, we present a fractal analysis procedure called‘‘Dimensional Interplay Analysis’’ and apply it to 14 paint-ings selected from Pollock’s catalog to represent the varietyof his poured work. Based on this research, we identify aprecise and highly distinguishable set of fractal characteris-tics. Secondly, we analyze 37 poured paintings generatedby students and also 14 poured paintings of unknown ori-gin (submitted by private collectors in the USA who believethe paintings to date from the Pollock era), and show thatnone of these paintings exhibit the set of fractal characteris-tics identified in Pollock’s paintings. We discuss the implica-tions of these results for authenticity research and considerthe extent to which this fractal analysis might be extendedto abstract and figurative paintings by other artists.

2. Fractal analysis

2.1. Fractal analysis of Pollock paintings

What is the identifying ‘‘hand’’ of Pollock? His pouredpatterns stand in sharp contrast to the straight lines, trian-gles and the wide range of other artificial shapes belongingto Euclidean geometry. Instead, his poured paintings arefrequently described as ‘‘organic’’, suggesting that theyallude to Nature. Since its introduction in the 1970s, fractalgeometry has experienced remarkable success in describingthe underlying patterns of many of nature’s objects, includ-ing coastlines, clouds, flames, lightning, trees and mountainprofiles (Barnsley, 1992; Gouyet, 1996; Mandelbrot, 1977).In contrast to the smoothness of Euclidean shapes, fractalsconsist of patterns that recur on finer and finer magnifica-tions, building up shapes of immense complexity. Given the‘‘organic’’ appearance of Pollock’s paintings (Potter, 1985),the first step towards identifying the ‘‘hand’’ of Pollock isto adopt the pattern analysis techniques used to identifyfractals in Nature’s scenery and apply the same processto his canvases. Nature’s fractals obey statistical self-simi-larity – the patterns observed at different magnifications,although not identical, are described by the same spatialstatistics (Mandelbrot, 1977). To detect statistical self-sim-ilarity, a scanned photographic image of the painting iscovered with a computer-generated mesh of identicalsquares. By analyzing which of the squares are ‘‘occupied’’(i.e., contains a part of the painted pattern) and which are

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‘‘empty’’, the statistical qualities of the pattern can be cal-culated. Reducing the square size in the mesh is equivalentto looking at the pattern at a finer magnification. Thus, inthis way, the painting’s statistical qualities can be com-pared at different magnifications.

As an example, we consider the analysis of Blue Poles:Number 11, 1952 (actual size 210.4 cm · 486.8 cm), forwhich we use a 42.0 cm · 97.4 cm high-resolution print ofthe painting as the source image. Our current analysis pro-cedures employ an HP Designjet 815mfp scanner, whichallows such an image to be scanned using a one stepprocess. We note, however, for the results reported in thisLetter, the scanner’s window size was limited to 30.0 cm ·42.0 cm, necessitating a two-step procedure as follows.First, a 30.0 m · 42.0 cm section of the print was scannedat 600dpi, creating a 24 bit color bitmap image with 9921pixels across the length. The analysis of this image exam-ines pattern sizes ranging from the smallest speck of paint(1 mm on the canvas) up to sizes matching the height of thecanvas (210.4 cm). Within this size range, the analysis isnot affected by image resolution limits, such as those asso-ciated with scanning procedures. For example, the pixelseparation after scanning corresponds to a physical sizeof 0.2 mm on the canvas, and thus the smallest painted pat-tern spans five pixels. A visual inspection confirms that thesmallest analyzed pattern (1 mm) is resolved. To analyzepatterns across size scales larger than the canvas heightof 210.4 cm, including those spanning the entire canvaslength of 486.8 cm, a smaller print (18.1 cm · 42.0 cm)was required in order to fit the painting’s full image withinthe length of the scanner window (42.0 cm). Employingthese two sets of analysis, we find that the largest observedfractal pattern is over one thousand times larger than thesmallest pattern (see below) (Taylor et al., 1999a,b). Thissize range is significantly larger than for typical observa-tions of fractals in other physical systems (where the largestpatterns are typically just 25 times larger than the smallestpattern) (Avnir et al., 1998). A consequence of observingthe fractal patterns over such a large size range is thatparameters that characterize the fractal statistics can bedetermined with accuracy.

2.2. Fractal dimensions of Pollock paintings

A crucial parameter for characterizing a fractal patternis the fractal dimension, D, and this quantifies the scalingrelationship between the patterns observed at differentmagnifications (Mandelbrot, 1977). For Euclidean shapes,dimension is a simple concept and is described by the famil-iar integer values. For a smooth line (containing no fractalstructure) D has a value of 1, while for a completely filledarea its value is 2. However, the repeating structure of afractal line generates a non-integer D value that lies inthe range between 1 and 2 (Schroeder, 1991). A fractal pat-tern’s D value can be determined by applying the well-established ‘‘box-counting’’ technique (Gouyet, 1996) tothe computer-generated mesh of squares discussed above.

Specifically, if N, the number of occupied squares (or‘‘boxes’’), is counted as a function of L, the square size,then for fractal behavior N(L) scales according to thepower law relationship N(L) � L�D (Gouyet, 1996; Man-delbrot, 1977). This power law generates the scale invariantproperties that are central to fractal geometry. The Dvalue, which charts this scale invariance, can be extractedfrom the gradient of the ‘‘scaling plot’’ of log(N) plottedagainst log(L). The standard deviation associated with fit-ting the data to the fractal scaling behavior is such thatD can be determined to an accuracy of two decimal places.The D value can be confirmed using a derivative analysis ofthe scaling plots (Taylor, 2002). To confirm the reliabilityof the whole (scanning and analysis) procedure, we gener-ate a set of computer-generated test patterns, includingEuclidean and fractal shapes. These test patterns areprinted at a resolution matching the source image of thepaintings, and then scanned and analyzed using the proce-dure outlined above. A comparison of the measured D val-ues with the known values for these test patterns confirmsthe accuracy of the procedure.

We adopt two commonly used magnification proceduresto construct the scaling plots and find them to be consis-tent. For the first procedure, the square size L is reducediteratively using the inverse expression L = H/n, where H

is the canvas size and n is the number of iterations(n = 1,2,3, . . .). For the second procedure, the exponentialexpression L = HC�n (where C is a selected magnificationfactor) is applied iteratively. The first procedure has theadvantage of generating a larger number of data points,while the second procedure reduces computation timeand produces equally spaced points across the resultinglog–log scaling plot. For both procedures, the validity ofthe counting procedure increases as L becomes smallerand the total number of boxes in the mesh is large enoughto provide reliable counting statistics (Gouyet, 1996). Intypical scaling plots of Pollock’s canvases, the large num-ber of boxes in the mesh ensures reliable counting statisticsacross the entire range of analyzed size scales. Consider, forexample, the mesh of squares covering the full image ofBlue Poles: Number 11, 1952. The smallest box size ofL = 1 mm corresponds to 16 million boxes across the can-vas. The coarsest length scale analyzed is set to ensure thatthe number of boxes does not become too limited at large L

values. For Blue Poles: Number 11, 1952, this is set atL = 30 cm, which corresponds to approximately one six-teenth of the canvas length and 112 boxes in the mesh.We note, however, that for some paintings well-definedfractal scaling behavior has been observed up to L valuescorresponding to 50 boxes in the mesh before significantscatter emerges in the scaling plot data.

A scaling plot resulting from a box-counting analysisperformed on Blue Poles: Number 11, 1952 is shown inFig. 3(a). The plot of N against L, on log–log (base 10)axes, is generated using the magnification expressionL = H C�n (where C = 1.1 and H = 1052 mm). This pro-duces 54 data points in the range of L shown, compared

Fig. 3. Box-counting analyses of (a) the aluminum layer of the paintingBlue Poles: Number 11, 1952 shown in Fig. 1; (b) the black layer of apainting generated by an art student; (c) the black layer of the painting ofunknown origin shown in Fig. 2.

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to over 1000 points generated from the equivalent inverseexpression L = H/n (Taylor et al., 1999b). For small sizesthe scaling plot follows one straight line (and hence oneD value) and then crosses over to another straight line ofa different gradient at the transition size marked LT. Wenote that this multi-fractal behavior, for which the scalingplots are characterized by two distinct scaling regimes, isexpected from the studies of the physical processes usedby Pollock (to be discussed in Section 2.4) and is alsoobserved in certain natural fractals (in particular, treeforms). We label the dimensions extracted from the twogradients the drip fractal dimension, DD, and the Levyfractal dimension, DL (see below for an explanation ofnomenclature). An iterative fitting procedure uses DD, DL

and LT as adjustable parameters to minimize the standarddeviation, sd, of the data from the two linear fit lines. Forthe fit shown in Fig. 3(a), the values of DD = 1.63,DL = 1.96 and LT = 1.8 cm produce sd = 0.020. At largerscales, approaching L � 50–100 cm (the size depends onthe painting), the painting ‘‘space fills’’ (i.e., all the boxesin the mesh are occupied) and, as a consequence, the gradi-ent shifts to D = 2. If a painting has a low DL value, thissecond transition to D = 2 will be clear in the scaling plots.This is not the case for paintings with high DL values (as isthe case for Fig. 3(b)), and the transition length is insteaddetermined by examining the box size below which emptyboxes start to occur.

We introduce the term ‘‘Dimensional Interplay Analy-sis’’ (DIA) to describe this technique of charting the inter-play of fractal parameters across distinct size scales. Thetechnique can be used to identify the precise trademarkbehavior of Pollock’s patterns (see Section 2.5). Of the 50Pollock paintings that have undergone a ‘‘box-counting’’analysis, we have applied this more detailed DIA to the fol-lowing paintings: Untitled (1945), Lucifer (1947), Full

Fathom Five (1947), Number 14, 1948 (1948), Figure(1948), Number 23, 1948 (1948), Number 8, 1949 (1949),Number 27, 1950 (1950), Number 32, 1950 (1950), Autumn

Rhythm: Number 30 (1950), Unknown (1950), Untitled

(1951), Blue Poles: Number 11, 1952 (1952) and Conver-

gence: Number 10, 1952 (1952) (on-going analysis). Thiscollection of paintings was selected to cover the diversityof Pollock’s poured work, ranging from one of his smallestpaintings (57.4 cm · 78.4 cm of Number 23, 1948) to one ofhis largest canvases (266.7 · 525.8 cm of Autumn Rhythm:Number 30, 1950), and paintings created using differentpaint media (enamel, aluminum, oil, ink and gouache)and supports (canvas, cardboard, paper and glass).

2.3. De-construction of Pollock paintings

Many of Pollock’s paintings feature a number of differ-ent colored layers of paint. These paintings are electroni-cally deconstructed into their constituent colored layersand a box-counting analysis is performed on each layer.For example, for Blue Poles: Number 11, 1952, six layerswere extracted for analysis (blue–black, aluminum, gray,light yellow, dark yellow and orange) and Fig. 3(a) showsthe box-counting analysis of the aluminum layer. The colorseparation is performed by identifying the RGB range (ona scale of 0–255 for each of the red, green and blue chan-nels) of the color variations within each layer and then fil-tering accordingly. For these multi-colored paintings, weemphasize that the layers are interactive in character – asthe trajectories of one layer were deposited they obscuredparts of the underlying layers’ patterns. The analyzed pat-terns therefore correspond to those seen by the observer ofthe complete painting rather than the patterns as they weredeposited. Of the 14 paintings listed above, 9 are multi-layer paintings (a proportion that is approximately repre-sentative of Pollock’s catalog of poured paintings). In total,

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32 layers have been extracted from the 14 paintings. Mea-sured sd values lie in the range 0.009 < sd < 0.027, with amean value of 0.019 and a standard deviation from thismean of 0.004. We note that sd decreases with canvas size:for example, the average sd value for the layers of Autumn

Rhythm: Number 30, 1950 (canvas area of 140,200 cm2) is0.020, compared to 0.010 for Number 23, 1948 (canvas areaof 4508 cm2).

Having separated the component colored layers andshown each of the analyzed layers to be fractal, some ofthe paintings have then been ‘‘re-constructed’’ by re-incor-porating the layers to build the complete pattern. As eachof the colored layers is re-incorporated, the two fractaldimensions, DD and DL, of the overall painting rise (Tay-lor, 2000, 2003). Thus the combined pattern of many colorshas higher fractal dimensions than any of the single layercolors. For example, the fractal dimensions of the alumi-num layer of Blue Poles: Number 11, 1952 are DD = 1.63and DL = 1.96, compared to the higher values ofDD = 1.72 and DL = 2 for the complete painting. The DD

value of the complete painting is a particularly sensitiveparameter for investigating the poured paintings. By ana-lyzing Pollock’s poured paintings over a decade (from1943 to 1952) we can use DD to quantify the evolution inhis fractal patterns, as shown in Fig. 4. Art theorists cate-gorize the evolution of Pollock’s pouring technique intothree phases (Varnedoe, 1998). In the ‘‘preliminary’’ phaseof 1943–1945, his initial efforts are characterized by low DD

values. An example is the fractal pattern of the paintingUntitled from 1945, which has a DD value of 1.12. Duringhis ‘‘transitional phase’’ from 1945 to 1947, he started toexperiment with the pouring technique and his DD valuesrose sharply (as indicated by the first gradient in Fig. 4).In his ‘‘classic’’ period of 1948–1952, he perfected his tech-nique and DD rose more gradually (second gradient inFig. 4) to the value of DD = 1.7. An example is Autumn

Rhythm: Number 30, 1950. During his classic period he also

Fig. 4. Fractal dimension DD plotted as a function of the year the patternwas painted by Pollock. The two lines serve as guides to the eye (see textfor details). Sections of Pollock’s paintings are shown as examples offractal patterns with different D values. Top to bottom: Untitled, 1950(DD = 1.89), the black layer of Autumn Rhythm: Number 30, 1950, 1950(DD = 1.66), the black layer of Untitled, 1945 (DD = 1.12).

painted Untitled, which has an even higher DD value of1.89. However, he immediately erased this pattern (it waspainted on glass), prompting the speculation that heregarded this painting as too complex and immediatelyscaled back to paintings with DD = 1.7. This suggests thathis 10 years of refining the pouring technique were moti-vated by a desire to generate fractal patterns withDD � 1.7. We note the potential use of the graph shownin Fig. 4 for establishing an approximate date for authenticPollock paintings.

2.4. The fractal generation process

Having identified the precise fractal characteristics ofPollock’s paintings, we now address whether these charac-teristics and the processes that generated them are uniqueto Pollock. Our previous fractal analysis of the film andphotographs of Pollock (recording the evolution of hispaintings at different stages of completion) reveals aremarkably systematic process (Taylor et al., 2003). Hestarted by painting localized islands of trajectories (approx-imately 50–100 cm in size) distributed across the canvas,followed by longer extended trajectories that joined theislands, submerging them in a dense fractal web of paint.This process was swift with the fractal dimensions of thepainting rising sharply in a time frame of several minutes(Taylor et al., 2003). Over an extended period of time(six months in the case of Blue Poles: Number 11, 1952),he would then deposit the extra layers of different coloredpaint on top of this initial ‘‘anchor’’ layer. During this finalstage he appeared to be fine-tuning the D values, with val-ues sometimes rising by less than 0.05. This deliberate,refined process raises the possibility that fractals are notan inevitable consequence of pouring paint but are insteadthe result of Pollock’s specific pouring technique (Tayloret al., 2003).

This speculation is supported by an analysis of the phys-ical processes used by Pollock to generate the two sets offractal patterns described by DD and DL. Studies of the filmand still photography (Taylor et al., 1999b; Taylor, 2000,2003) show that the small scale patterns quantified by DD

were predominantly generated by the pouring process –the dynamics of the fluid paint as it fell though the airand interacted with the canvas. The larger scale patternsdescribed by DL were predominantly shaped by Pollock’smotion. The potential of these two processes to generatefractals depends critically on parameters relating to Pol-lock’s painting technique (Taylor et al., 1999b; Taylor,2000, 2003). It is well established that falling fluid candecompose into a sequence of fractal droplets (Shi et al.,1994). In Pollock’s case, formation of fractal dropletswould depend on the paint viscosity and the manner inwhich he ‘‘launched’’ the paint from his painting imple-ment. If, for example, the paint was too fluid, or the dropdistance was too short, then fractal drops would not form.Similarly, fractals permeate human motion only under spe-cific conditions. Our preliminary research indicates that

Fig. 5. An example of a poured painting generated by a student(61.6 cm · 91.5 cm).

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human motion displays characteristics of fractal ‘‘Levyflights’’ (Taylor, 2004) when people restore their balance.This suggests that perhaps Pollock’s motions were per-formed in a state of controlled ‘‘off-balance’’, and that hedeliberately tuned into the fractal behavior of this underly-ing physiological process. This motion produced fractaltrajectories that linked together the islands in the underly-ing anchor layer. In general, Pollock did not paint trajecto-ries longer than the island size of approximately 100 cm,and consequently at larger scales the painting is non-fractalwith D = 2.

In summary, the emerging science of his painting pro-cess suggests that Pollock’s fractals are the product of theparameter conditions chosen by the artist. This is con-firmed by our analysis of the patterns of paint found onPollock’s studio floor. These patterns, created by the paintthat missed the canvas, are not fractal, emphasizing thatPollock’s fractals were a direct consequence of the way thathe aimed the paint at the canvas – the fractals were a prod-uct of Pollock’s compositional technique. The concept thatpoured paintings are not inevitably fractal is further sup-ported by our studies in which a chaotic pendulum wasemployed to pour paint onto a horizontal canvas (Taylor,1998, 2003). When the chaos in the swinging motion wassuppressed, the resulting poured paintings were devoid offractal content (Taylor, 2002).

2.5. Fractal analysis of poured paintings not attributed to

Pollock

Fractal patterns are therefore the product of the specifictechnique Pollock used to pour paint and all of the ana-lyzed poured paintings have this fractal composition. Infact, the ‘‘hand’’ of Pollock is more specific than fractality– through our empirical study, we can identify the specifictrademark features of Pollock’s abstract paintings. Thesetrademarks combine to have a profound impact on thevisual character of his work and are summarized as fol-lows: (1) Pollock paintings are composed of two sets offractal patterns (generated by the pouring process and hisbody motions). (2) These fractal patterns occur over dis-tinct length scales. In particular, although the transitionallength scale LT varies between Pollock paintings (depend-ing on factors such as canvas size), the transition satisfiesLT P 1 cm. (3) Each set of fractal patterns is well describedby a fractal dimension extracted from the gradient of thelog–log scaling plot. (4) The condition DL > DD is alwayssatisfied. (5) The quality of the fits of the scaling plot datato the above characteristics is high. Measured sd valuesnever exceed 0.027, with lower sd values down to 0.009measured for the smallest paintings. (6) If the painting iscomposed of a number of colored layers, each of the ana-lyzed layers satisfies the previous five criteria.

To examine the extent to which this set of fractal char-acteristics is unique to Pollock, 37 undergraduate studentsfrom the University of Oregon were asked to generateabstract paintings using the pouring technique. The paint-

ing parameters were chosen to match Pollock’s more sim-ple, black and white poured paintings such as Number 23,1948 (57.5 cm · 78.4 cm). The students were asked to painta single black layer on an area measuring 61.6 cm ·91.5 cm. The box-counting analysis revealed that none ofthe 37 paintings matched all 6 of the required criteria. Inthe attempt to match to the criteria, the computer fits thescaling plot data to two straight lines (one for low L valuesand the other for high values), as required by criterion 1.This fitting procedure uses the LT value as a free variableto minimize the standard deviation of the fit lines to thedata. If the LT value obtained by the fitting procedure lieswithin the accepted size range (criterion 2), then the othercriteria are assessed. If the LT value fails to lie within therequired range, the painting is given a ‘‘second chance’’,for which the fitting procedure is repeated usingLT = 3.0 cm (the average LT value for all analyzed Pollockpaintings). Using this ‘‘forced fit’’, the remaining criteriaare then assessed. All of the 37 paintings required theforced fit procedure. An example painting (labeled Drip-

fest35) is shown in Fig. 5. Despite any superficial similari-ties with Pollock’s work, this painting does not match therequired criteria listed above. The ‘‘forced-fit’’ analysis isshown in Fig. 3(b). To facilitate a comparison of qualityof fit between the paintings produced by Pollock and thestudents, the fitting procedure is assessed over a magnifica-tion range of 1 mm <L < 10 cm and uses the same magni-fication factor (C = 1.1) to ensure an analogous numberof data points across this range of L. For the fit shownin Fig. 3(b), sd = 0.053, compared to 0.010 for Pollock’sequivalent Number 23, 1948. For the 37 paintings, the sdvalues of the fits fall outside the range observed for Pollockpaintings. Their sd values are the range 0.030 < sd < 0.053,with a mean value of 0.037 and a standard deviation fromthis mean of 0.006.

For poured paintings of unknown origin, it is informa-tive to perform a fractal analysis to determine if the abovecriteria are satisfied. To date, we have analyzed 14 pouredpaintings supplied to us by private collectors. None of these

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paintings matched the required six criteria. Out of the 49layers extracted from the 14 paintings, 18 used the ‘‘free-fit’’ procedure. An example is the black layer of the paint-ing shown in Fig. 2 (labeled Unknown12). The painting’sscaling plot is shown in Fig. 3(c) and the fit is quantifiedby sd = 0.038. For the 14 paintings, the sd values lie inthe range of 0.031 < sd < 0.072. We note that the largersd values exhibited by both the student paintings and thepaintings of unknown origin quantify clear deviations fromthe established Pollock fractal characteristics (i.e., the sixcriteria). These deviations are clearly evident in all of thescaling plots – as demonstrated, for example, in Fig. 3(b)and (c). The manner in which the data deviates from Pol-lock scaling behavior varies between paintings and herewe describe a few common categories: (1) the scaling plotsfail to display scale invariance across any length scales.Fig. 3(c) is an example of this behavior. (2) The scalingplots exhibit well-defined DL behavior but fail to condenseonto a well-defined DD behavior (this commonly results ina forced fit procedure because the free fit produces an LT

value well below L = 1 cm). Fig. 3(b) is an example of thisbehavior. (3) The scaling plots display D = 2 across manylength scales, indicative of a ‘‘space-filled’’ canvas pro-duced by excessive paint deposition. (4) The scaling plotsdisplay DD = 1, indicative of overly smooth paint trajecto-ries that lacked ‘‘splatter’’ in the pouring process.

3. Conclusions

We conclude by emphasizing that our initial results pre-sented in this Letter are part of an on-going research pro-ject. Nevertheless, the results obtained to date demonstratea 100% success rate in terms of identifying Pollock’s spe-cific form of fractal behavior. This is in contrast to the100% failure rate of the non-Pollock paintings of knownorigin – none of these paintings satisfy the outlined criteria.These initial results demonstrate the potential of computersto detect the trademark characteristics of Pollock’s patternsand to use fractal analysis as an authenticity tool. The100% failure rate of the paintings of unknown origin sentto us by private collectors is consistent with expectations– non-Pollock poured paintings are believed to vastlyout-number undiscovered real Pollock paintings (for exam-ple, in the 17 years between publication of the Pollock Cat-alogue Raisonne and its supplement (O’Connor and Thaw,1978; O’Connor, 1995), only six new Pollock paintingswere authenticated). In future work, the fractal analysispresented here will be particularly effective when combinedwith the traditional authenticity tools outlined earlier andother fractal analysis techniques that have recently beenapplied to Pollock’s poured paintings (Mureika et al.,2005; Mureika, 2005). The latter research concentrates onthe DL regime, and includes a multi-fractal spectral analy-sis (measuring a range of dimensions that build on the‘‘box-counting’’ DL) and also an investigation of the DL

behavior of the fractal edge patterns at the boundariesbetween the paint and canvas. Future collaborations will

also focus on wavelet analysis (Lyu et al., 2004) and patternconnectivity analysis (Martin et al., 2004).

Finally, we note that the impact of fractal analysis as anauthenticity tool is not restricted to Pollock’s pouredworks. For abstract artists who painted non-fractal pat-terns, fractal analysis could be used to detect the trademarkcharacteristics of how their patterns deviate from fractalscale invariance. Furthermore, fractal analysis can beapplied to both abstract and figurative paintings. Fractaland other forms of scale analysis have been used by otherresearchers to investigate artists as diverse as early Chinesefigurative painters (Voss, 1998), Vincent Van Gogh (Herikand Postma, 2000) and Pieter Bruegel the Elder (Lyu et al.,2004). The application of fractal analysis is not restricted tothe painted image of an artwork – for example, stress frac-tures are fractal and therefore fractal analysis can be usedto investigate the cracks that form in the paint layers ofageing paintings. More generally, we anticipate that fractalanalysis will be integrated with other forms of computer-analysis, with the goal of authenticating paintings using awide spectrum of quantifiable parameters. These parame-ters will extend beyond the ‘‘form’’ of the painting toinclude information relating to color, physical size andmaterials. In particular, the latter will be collected usingan array of scientific tools (which includes X-ray, ultra-violet and infrared radiation techniques, optical and elec-tron-beam microscopy (McCrone, 2001) and forensicstechniques such as fingerprint and DNA analysis). Ulti-mately, art scholars and connoisseurs will be able tocombine their knowledge with a database of scientificinformation involving the entire catalog of known paint-ings by the artist. In this sense, the research in this Letter,along with other projects such as the Rembrandt ResearchProject (Bruyn et al., 1990) and Authentic (Heingartner,2004), represent the future direction of authenticity workand signal the growing interplay between art and science.

Acknowledgements

R.P.T is a Research Corporation Cottrell Scholar.T.P.M. and C.A.M are National Science FoundationIGERT fellows. R.P.T. thanks F.W. O’Connor (authorof the Pollock Catalogue Raisonne), J. Coddington (Con-servator, Museum of Modern Art, New York), H. Harri-son (Director, Pollock–Krasner Study Center) and S.Flescher (Executive Director, International Foundationfor Art Research) for useful discussions during this pro-ject’s development.

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