To quantify the fractal content of Pollock's drip paintings, such as Alchemy (Fig. 1), we used the well-established ‘box-counting’ method2 to calculate the fractal dimension D. We cover the scanned photograph of a Pollock painting with a computer-generated mesh of identical squares. The number of squares N (L) that contain part of the painted pattern is then counted; this is repeated as the size, L, of the squares in the mesh is reduced. The largest size of square is chosen to match the canvas size (L ≈ 2.5 m) and the smallest is chosen to match the finest paintwork (L ≈ 1 mm). For fractal behaviour, N (L) scales according to N (L)L D, where 1 < D < 2. The D values are extracted from the gradient of a graph of log N (L) plotted against log L. This fractal analysis reveals two distinct D values occurring over the ranges 1 mm < L < 5 cm and 5 cm < L < 2.5 m. Our analysis of a film of Pollock while painting shows that the fractal patterns occurring over the lower range are determined by the dripping process, whereas the fractal patterns across the higher range are shaped by his motions around the canvas.

Figure 1: Alchemy, painted by Jackson Pollock in 1947.
figure 1

Drip paintings of this period are characterized by fractal dimensions close to 1.5. Reproduced by permission of ARS, NY and DACS, London, 1999.

Our analysis shows that Pollock refined his dripping technique: the fractal dimensions increased steadily through the years, from close to 1 in 1943 to 1.72 in 1952. Because D follows such a distinct evolution with time, the fractal analysis could be used as a quantitative, objective technique both to validate and date Pollock's drip paintings. The change in D reflects a dramatic evolution in visual character. His initial drip paintings of 1943 consisted of a single layer of paint trajectories that occupied only 20% of the 0.35-m2 canvas area; by 1952 he was painting multiple layers of trajectories that covered over 90% of his 9.96-m2 canvas. It is important that Pollock introduced his fractals systematically: the initial fractal layer essentially determined D by acting as an anchor layer for the subsequent fractal layers, which then fine-tuned the value of D.