“Number 1A”, Jackson Pollock, 1948
Both these paintings are at the Museum of Modern Art in New York. Mondarin’s painting (first one above) is simple, clean and structured. Pollock’s on the other hand, is tangled, messy and complex. Which one do you like better?
Mondarin spent weeks contemplating the precise arrangement of his lines. He was considered an intellectual of his time, who wrote thoughtful essays about his paintings. Pollock, on the other hand, simply rolled out his canvas on the floor of his studio and ran around dripping household paint on it from a stick, in what seemed to be a completely arbitrary process to observers (a video documenting his process below). Pollock was mostly drunk and didn’t bother justifying his work. Once he was quoted as saying “my concerns are with the rhythms of nature”.
Pollock's Process
In 1999, a U. Oregon physicist named Richard Taylor discovered that Pollock’s paintings are indeed fractal shapes[1]. It turns out Pollock was creating fractal patterns with his seemingly arbitrary painting style.
Fractals, of course, are the selfsimilar patterns found everywhere in nature. These patterns repeat themselves at different levels of magnification. Think cauliflower, tree branches, snails, lungs, snowflakes.
Romanesco Broccoli
Mandelbrot[2] showed in 1975 that fractals can be expressed as selfsimilar mathematical functions. For e.g. the Julia set
$$z = z^2 + c$$
produces the following plot on the realimaginary plane when c takes a complex value $$c = 0.8 + 0.156i$$
Example Julia Set
Fractals occupy the bizzare world of fractional dimensional values, or the “roughness” of a fractal. In our familiar world of Euclidian geometry, points have a dimensional value D = 0; lines have a dimensional value D = 1; surfaces have a dimensional value D = 2 and solids have a dimensional value D = 3. Fractals have dimensional values that fall in between these whole numbers, i.e. fractions, sometimes even irrational numbers (!).
For example, the cantor set (below) has a fractal dimension D = 0.63, which means that its behavior is somehwere between a point and a line. The Koch curve at D = 1.26 behaves more like a line, whereas a Sierpinski carpet at D = 1.89 behaves more like a surface. The Menger Sponge, at D = 2.73, is 'almost' a solid.
Cantor Set (D = 0.63)
Koch Curve (D=1.26)
Sierpinski Carpet (D = 1.89)
The most common way of measuring the fractal dimension (also called fractal density) of an image is the “Box Counting” method. Roughly, the value of D can be obtained by dividing up the image into smaller and smaller squares, comparing N(L) the number of occupied squares to L the size of the square. For fractal behavior, N(L) should scale according to the relationship
$$N(L) \backsim L ^ {D}$$
Natural scenery, too, come in varying values of D. Typical fractal dimension values found in natural scenery summarized in the table below. Now, Taylor surveyed 120 people to see what patterns they preferred, 113 of them chose fractals over nonfractals. Even in fractals, it turns out that people prefer images with a dimensional value of around 1.3.
It's remarkable that Pollock was creating fractal patterns in his paintings long before fractal geometry was officially “discovered” by Mandelbrot in 1975. Taylor also analyzed several Pollock paintings for D values and discovered that as years went by, the complexity, i.e. fractal density of his paintings went up as well. Here is a plot of D over time for Pollock’s paintings[5]:
Natural Pattern

Fractal Dimension D

Galaxies (modeled)

1.05 – 1.25

Cracks in ductile materials

1.25

Geothermal rock patterns

1.251.55

Woody plants and trees

1.281.90

Waves

1.3

Clouds

1.3 – 1.33

Snow flakes

1.7

Retinal blood vessels

1.7

Electrical Discharges

1.75

Common D values in Natural Scenery
Fractal Density vs. Aesthetic Preference
It's remarkable that Pollock was creating fractal patterns in his paintings long before fractal geometry was officially “discovered” by Mandelbrot in 1975. Taylor also analyzed several Pollock paintings for D values and discovered that as years went by, the complexity, i.e. fractal density of his paintings went up as well. Here is a plot of D over time for Pollock’s paintings[5]:
Fractal Density in Pollock's paintings plotted against the year in which they were painted
Is this the secret to the aesthetics of Pollock’s paintings?
The other interesting question is, are we conditioned evolutionarily to appreciate images of a particular fractal density, because that’s what we see all around us?
The other interesting question is, are we conditioned evolutionarily to appreciate images of a particular fractal density, because that’s what we see all around us?
Also, how do we expand this intuition to three dimensional objects? Do humans have an aesthetic preference for fractals in three dimensions as well i.e. solids?
Only one way to find out. In the next few months, I'm hoping to repeat Taylor's experiment for for fractals less than 2 dimensions. At the same time, I will attempt to design an experiment to test aesthetic preferences for solids as well. More to come!
References
1. Fractal Expressionism  Where art meets science, Richard Taylor, Santa Fe Institute Feb 2002
2. Fractal Geometry of Nature, Benoit Mandelbrot 1982
3. Pollock at the MoMA http://www.moma.org/explore/inside_out/2013/04/17/momasjacksonpollockconservationprojectinsightintotheartistsprocess/
4. Mondarin at the MoMA
http://www.moma.org/collection/artist.php?artist_id=4057
5. Perceptual and Physiological Responses to Jackson Pollock's Fractals, Richard P. Taylor, Branka Spehar, Paul Van Donkelaar, Caroline M. Hagerhall, Hum. Neurosci. 2011; 5: 60.
5. Perceptual and Physiological Responses to Jackson Pollock's Fractals, Richard P. Taylor, Branka Spehar, Paul Van Donkelaar, Caroline M. Hagerhall, Hum. Neurosci. 2011; 5: 60.
Published online 2011 June 22.
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