Fractal Artwork: Dimension and Complexity as a Guide for Aesthetics

Jackson Pollock is an artist famously known for his technique of paint dripping to create various works of art, using a pouring technique to create designs as opposed to more Euclidean shapes produced by brush strokes.  Due to this method of painting and the results it produces, Pollock’s works can be described to be fractal images, with his style being “Fractal Expressionism” [1].  Characterizing these fractal images comes in the form of fractal dimension, having a value that ranges between 1 and 2. A dimension of 1 would have no fractals, and values closer to 1 tend to be sparser; on the other hand, a dimension of 2 would be completely filled, and values near this would be more intricate and complex [2].

Determining the fractal dimension of a piece is done by the box counting method.   First, a computer generated mesh consisting of identical squares, or boxes, is placed on a digital image of the painting.  Next, scaling qualities of the fractal pattern are determined by calculating the proportion of filled squares to empty ones.  The number of occupied squares, given by N(L), is compared to the width L, where N(L) is a function of this width.  N(L) scales according to the power law that N(L) is asymptotic to L^-D, where D is the dimension.  Then, a scaling plot is created between –logN(L) and logL.  Finally, if the painting is of fractal pattern, this plot will produce a straight line, where the dimension D is the gradient of this straight line [2].

Using this method, we can characterize the fractal art of Pollock, and any other forms of fractal images.  Analyzing Pollock’s work in this way, mathematicians were able to identify “periods” in his work, in which his artwork would stay near a given value of fractal dimension during a time period of a few years [1]. In fact, the fractal analysis of characteristics in Pollock’s work shows that the fractals produced are not simply a consequence of paint being poured, but moreover by his own technique, involving deliberate and specific pouring, along with body motions.  Furthermore, Pollock’s work in this analysis meets 6 specific criteria. Some of these include having two fractal sets within his pieces, due to his techniques, as well as fractal patterns occurring over distinct length scales [1].  These criteria can then be used to identify Pollock’s work, as other artists trying to replicate his images will not meet the criteria when analyzed.

Figure 1. A graph of the change in fractal dimensions Pollock exhibits over time [1]

Aside from Pollock’s work, fractal dimension characteristics can also be used as a means to quantify aesthetics.  As stated previously, pictures range in fractal dimension value from 1 to 2; determining which values are considered aesthetically pleasing to viewers could help ascribe a number to aesthetic works.  One study conducted used participants to give preference when shown a number of fractals, each with different dimension [2].  This particular study had shown that participants preferred a fractal dimension of around 1.3 to 1.5, which was consistent throughout multiple types of fractals, including natural images and parts of Pollock’s works. It is important to note that a control was held, showing that preference was not dependent density, but over complexity alone.  A different study compared these various types of fractals, which found that natural images were most often considered the most beautiful, as well as having the highest fractal dimensions [3].  Finally, one last study had asked participants to rate sets of images, each having one of two fractal dimensions and one of two Lyapunov exponents [4].  This exponent quantifies the unpredictability of the process used to generate fractal images; in this case, the range falls between 0.01 and 0.84 bits per iteration, with the higher the number producing a more chaotic image.  This study found that the mean preference was 1.26 in fractal dimension, and 0.37 for the Lyapunov exponent, reflective of many natural images.

Figure 2. Fractal dimension preference graphs for different sets of fractals [2]

These various studies all attempt to affix a number as to what is deemed to be aesthetic.  By isolating a variable such as fractal dimension and running tests to see what is preferred, we can begin to find what constitutes an aesthetic fractal image.  However, these studies are still rooted in subjectivity, as well as being confined to very specific images that can be analyzed as data.  Therefore, quantifying art in this fashion does not define what art truly is.  Regardless of this fact, it does help to bridge a gap and allow some form of rating to be applied to a mostly qualitative field.

References:

[1]  Taylor, Robert P., et al. “Authenticating Pollock paintings using fractal geometry.” Pattern Recognition Letters 28.6 (2007): 695-702.

[2]  Spehar, Branka, et al. “Universal aesthetic of fractals.” Computers & Graphics 27.5 (2003): 813-820.

[3]  Forsythe, Alex, et al. “Predicting beauty: fractal dimension and visual complexity in art.” British journal of psychology 102.1 (2011): 49-70.

[4]  Aks, Deborah J., and Julien C. Sprott. “Quantifying aesthetic preference for chaotic patterns.” Empirical studies of the arts 14.1 (1996): 1-16.

[5]  Jones-Smith, Katherine, and Harsh Mathur. “Fractal Analysis: Revisiting Pollock’s Drip Paintings.” Nature 444.7119 (2006): E9-E10. Academic Search Complete. Web. 30 Nov. 2016.