In a blog back in January, I referenced a talk given by David Deutsch in which he made the argument that, while empiricism has been the basis of science, empiricism alone is inadequate because scientific theories explain the seen in terms of the unseen.
What we see, in all these cases, bears no resemblance to the reality that we conclude is responsible – only a long chain of theoretical reasoning and interpretation connects them.
This ‘theoretical reasoning and interpretation’ has its own structure, a mostly mathematical one. I am becoming more and more intrigued by the analogs that can be found among the structures in nature, the structures we see in the underlying action of sensory mechanisms and even in the structure or shape taken by the observable behavior of an organism because, not surprisingly, mathematics touches all of them. These analogous structures come to mind today because my attention was brought to what has been discussed, for a number of years now, about the fractal patterns in paintings by Jackson Pollock, one of the pioneers of abstract expressionism.
In an article that appeared in Physics World magazine in October, 1999, physicist Richard P. Taylor argued that the patterns in paintings produced by Jackson Pollock in the late 1940’s and early 1950’s are fractal. Then, in 2011, Taylor co-authored a paper that appeared in the journal Frontiers in Human Neuroscience entitled Perceptual and Physiological Responses to Jackson Pollock’s Fractals.
This paper is a more thorough analysis of Taylor’s idea with a much broader narrative. Alongside an image of the Long Island house where Pollock lived when he began his ‘pouring’ technique the authors note:
In contrast to his previous urban life in Manhattan, Pollock perfected his pouring technique surrounded by the complex patterns of nature. Right: Trees are an example of a natural fractal object. Although the patterns observed at different magnifications don’t repeat exactly, analysis shows them to have the same statistical qualities.
The suggestion is that Pollock, inspired by what he saw around him, had an insight about what was there, which he then reproduced for us. And this is what the artist does. But if Pollack was trying to reproduce the fractal nature of what he perceived, we have here another instance of an individual ‘seeing’ a deep and complex pattern without the use of any analytic tools (like mathematics). And this inevitably tells me something about mathematics.
The paper also makes some observations of Pollock’s physical action when painting, and of the evolution of his paintings (which suggests a clear directedness in his efforts).
The question of how Pollock combined the blobs into an integrated, multi-colored visual fractal led us to investigate his painting technique in detail. We described Pollock’s style as “Fractal Expressionism” (Taylor et al., 1999b; Taylor, 2011) to distinguish it from computer-generated fractal art. Fractal Expressionism indicates an ability to generate and manipulate fractal patterns directly. In many ways, this ability to paint such complex patterns represents the limits of human capabilities. Our analysis of film footage taken at his peak in 1950 reveals a remarkably systematic process (Taylor et al., 2002). He started by painting localized islands of trajectories distributed across the canvas, followed by longer extended trajectories that joined the islands, gradually submerging them in a dense fractal web of paint. This process was very swift with the fractal dimension rising sharply…
…he perfected this technique over 10years. Art theorists categorize the evolution of Pollock’s pouring technique into three phases (Varnedoe and Karmel, 1998). In the “preliminary” phase of 1943–1945, his initial efforts were characterized by low D values. An example is the fractal pattern of the painting Untitled from 1945, which has a D value of 1.10. During his “transitional phase” from 1945 to 1947, he started to experiment with the pouring technique and his D values rose sharply. In his “classic” period of 1948–1952, he perfected his technique and D values rose more gradually to the value of D=1.7. During his classic period he also painted Untitled which has an even higher D value of 1.89. However, he immediately erased this pattern (it was painted on glass), prompting the speculation that he regarded this painting as too complex and immediately scaled back to paintings with D=1.7. This suggests that his 10years of refining the pouring technique were motivated by a desire to generate fractal patterns with D~1.7.
The D-value or dimension of a fractal is a measure of the amount of fine structure in the fractal pattern.
The paper also addresses the question of why or whether the D-value that Pollock seems to move toward in his paintings has particular aesthetic appeal. Our eyes move in fractal patterns and I have blogged about this before. Experimental collaborations between psychologists and neuroscientists have found that images matching the fractal dimension of the eye’s searching movement are ones that are most aesthetically pleasing.
But the paths created by the motion of our eyes look very much like the path of a flying insect searching of food. And studies seem to indicate that the brain forages memory in much the same way.
This one mathematical thought seems to run through our existence on multiple levels. The ‘monstrous functions,’ used in the 19th century to demonstrate the break between mathematics and visible reality are now the fractals we use to describe physical and biological complexity. But it is mathematics that has characterized the pattern in such a way as to be able to see it. And we see it outside of us, inside of us – in the trees and the way we remember, in the structure of our lungs and the movement of our eyes (which tells us something about mathematics).