## Fractals and Jackson Pollock

My first paper appeared in Nature and debunked claims that the drip paintings of Jackson Pollock can be considered fractal [1]. In the late 1990s a group of physicists claimed to have identified a unique set of fractal characteristics present in Pollock's paintings, and that fractal analysis could be used as an authentication tool for his work [2,3].  They attributed the fractal characteristics to an inferred chaotic motion undertaken by Pollock as he painted. Pollock died in 1956 and there is no way to observe his motion while he painted, other than a few minutes of film footage [4,5]. In this footage and other sources, Pollock famously says "I can control the flow of the paint. There is no accident." [5]  and ``No chaos, damn it.'' [6]. He also said of his creative process: "I want to express my feelings, rather than illustrate them" [5].  I think these statements should be borne in mind when considering whether a mathematical characterization of his paintings is warranted, necessary, or desirable.

The inconsistencies that follow from the 'hypothesis of fractal expressionism' mostly stem from the limited range of length scales available in even the largest of Pollock's paintings. The largest of his paintings are 1-2 m in length/height, while the smallest speck of paint is of order 1mm. This range represents three orders of magnitude over which fractal dimensions could be in principle be determined. However, this range is used to calculate two purportedly independent fractal dimensions (one dominating on large length scales and one dominating on small), thus only about 1.5 orders of magnitude are used to determine each fractal dimension in the largest paintings.  Though there are many published accounts of physical systems that demonstrate fractal behavior over a smaller range than three orders of magnitude, (i) this doesn't make any given claim of fractal behavior robust or obviate the need for data which demonstrate the scaling properties, and (ii) both sides of a prominent debate agree that less than three orders of magnitude is an insufficient range and labelling such systems as fractal represents, according to Mandlebrot "unfortunate side effects of enthusiasm, imperfectly controlled by refereeing, for a new tool that was (incorrectly) perceived as simple.”  [7].

The extent to which Jackson Pollock's paintings can be considered fractal is a very limited one, and to this same extent many other images can be considered fractal as well. Crude sketches made by me in Adobe Photoshop, which involved no chaotic motion whatsoever and do not even resemble drip paintings, are nonetheless authentic Pollocks according to a computer using the criteria of fractal expressionism. My Untitled 5, for example, has all of the "fractal" characteristics of a Pollock painting. We also have deliberately fraudulent drip paintings made by undergraduates at Case Western Reserve University in 2007 that satisfy the same criteria and are authentic Pollock paintings made in 1948, according to the fractal authentication criteria. Meanwhile known, authentic Pollocks fail to be authentic [8].

The question of whether Pollock was actually undergoing chaotic motion is rendered moot as a consequence of the limited range. There are many types of chaotic motion, and some leave fractal trails. The Levy type motion assumed in [2] is a scale-invariant process, where the number of steps as a function of step-size follows a power law distribution. There is no characteristic length scale present in this type of motion, and the trajectory is a fractal [9]. On the other hand, a Gaussian random walk is a type of chaotic motion in which the number of steps is Gaussian distributed about the mean step size; the mean step size is the characteristic length scale in this type of motion. Though Gaussian random walks are chaotic, the trajectory left by a particle undergoing a this type of motion is not fractal, it is space-filling and has dimension 2 when observed over sufficiently long times. We can get a glimpse of Pollock's stride as he paints in [4]; a man of a certain height moving around a canvas of a fixed size would almost certainly have a few characteristic length scales associated with his motion.  Nonetheless the premise for the hypothesis of fractal expressionism is that Pollock underwent the scale-invariant Levy type motion as he painted.

There are a few interesting questions that we uncovered in the context of fractal analysis of Pollock's paintings. In fractal expressionism, one supposes that every layer of every Pollock painting, as well as the composite image, is fractal. It is not clear that overlapping fractals would be fractal themselves, and we found that in the case of Cantor dusts this is not true: the unobscured part of a lower layer and the composite image are not fractal, even when the two constituent layers are perfect mathematical fractals (Cantor dusts). This is one problem faced by fractal expressionism that does not owe its nature to the limited range. Some other interesting mathematical findings are discussed in [1] and [8].

References:
[1] Jones-Smith, Katherine, and Harsh Mathur. "Fractal analysis: revisiting Pollock's drip paintings." Nature 444.7119 (2006): E9-E10. (See also Media section of my homepage.)
[2] Taylor, Richard P., Adam P. Micolich, and David Jonas. "Fractal analysis of Pollock's drip paintings." Nature 399.6735 (1999): 422-422.
[3] Taylor, Richard P. "Order in Pollock's chaos." Scientific American 287.6 (2002): 84-89.
[4 ] https://www.sfmoma.org/watch/jackson-pollock-paintings-have-a-life-of-their-own/
[5] Hans Namuth films.
[6] Pollock's response to Time Magazine's 1950 article entitled "Chaos, Damn It." He wired the message "NO CHAOS DAMN IT."  See "Jackson Pollock: Key Interviews, Articles, and Reviews, by Pepe Karmel and Jackson Pollock. Museum of Modern Art Publishing, 2001.  See also http://www.moma.org/explore/conservation/pollock/interview1.html
[7] Avnir, David, et al. "Is the geometry of nature fractal?." Science 279.5347 (1998): 39-40.
[8] Jones-Smith, Katherine, Harsh Mathur, and Lawrence M. Krauss. "Drip paintings and fractal analysis." Physical Review E 79.4 (2009): 046111.
[9] Mandlebrot, Benoit. The Fractal Geometry of Nature. W.H. Freeman and Company, New York. 1977.