\graphicspath{{PollockDimension/}}
\section{Jackson Pollock's Fractal Paintings}
Jackson Pollock began his work on drip paintings in 1942. Today he is considered by many to be the most famous of abstract artists. He created many of his works by dripping paint onto large canvases. 
His technique is recognized by many as an important advancement in the development of modern art; however, the artistic value of the resulting works is a topic for debate. To the casual observer it may appear that Pollock was simply applying paint in random dribbles,but researchers have recently examined his paintings and discovered the fractal property of self-similarity rather than unorganized, random patterns. One researcher in particular, Richard Taylor, has spent a lot of time exploring the fractal nature of Pollock's work. Taylor is a professor of physics at the University of Oregon. He also has a master's degree in art theory from the University of New South Wales with a focus on Jackson Pollock's work. Two main observations come out of Taylor's work: Pollock's patterns are fractal in a way that reflects nature, and the fractal dimensions of Pollock's paintings increased during his career.

Both images used in this section are from Professor Taylor's web page. \footnote{http://materialscience.uoregon.edu/taylor/art/fractal\_taylor.html}
In an effort to quantify the fractal nature of Pollock's drip paintings, Taylor and his team used the box-counting method to determine the fractal dimensions of the works. This is the same box-counting method described in Section \ref{sec:dimension:boxcounting} on page \pageref{sec:dimension:boxcounting}.
 They began by overlaying a computer-generated grid on top of the painting and counting the number of squares that contain part of the painted image. This process is repeated with various grid sizes. Next a plot of the number of squares versus the grid size on a $log$/$log$ axis provides a graphical representation of the fractal dimension.  Taylor discovered that, for much of Pollock's work, this graph contained two distinct linear regions. This equates to two separate values for the fractal dimension, dependent on the resolution of the computer generated grid. For square size, $r$, in the range $1mm < r < 5cm$, one value was found, and for $5cm < r < 2.5m$ another was discovered. 
 Figure \ref{fig:pollockloglog} shows a box-counting plot of one of Pollock's works. The two different slopes of the line indicate that there are two different fractal dimensions in the piece; one at a large scale and another at a small scale.


\begin{figure}
\centering
\includegraphics{autumnrhythmwhole}
\caption{Autumn Rhythm. An example of Pollock's work. The right side
shows the whole work while the left side shows just the black anchor layer.}
\label{fig:autumnrhythm}
\end{figure}


\begin{figure}
\centering
\includegraphics{pollockloglog}
\caption{Box-counting analysis of Pollock's work showing the presence of
two different fractal dimensions in one painting.}
\label{fig:pollockloglog}
\end{figure}

The fractal dimension computed for the smaller grid size range is attributed to the dripping process itself. Factors such as paint viscosity, paint absorption into the canvas, etc., are attributed to this chaotic behavior. The fractal dimension of the larger square size range is thought to be a direct product of the painter's motion. In other words, even though Pollock did not have any technical knowledge of chaos theory or fractals (these fields of study did not even exist when he was painting), it was still his intent to create fractal patterns. This is supported by Pollock saying that his concerns were with ``the rhythms of nature''. In Taylor's words, ``He described Nature directly. Rather than mimicking Nature, he adopted its language - fractals - to build his own patterns.''[Taylor 1999]

When Pollock began making drip paintings in 1943 they consisted of a single layer of paint occupying only about 20\% of the canvas area. His later paintings, in contrast, contained multiple layers of paint trajectories that covered over 90\% of the canvas. The fractal dimension also increased as he continued his work, starting at values close to 1 and getting as high as 1.72 in 1952. This change in fractal dimension is a reflection of a dramatic change in visual character. Taylor believes that since the fractal dimension increases distinctly with time, fractal analysis could provide a quantitative way to date Pollock's paintings, and to detect forgeries.

It seems that fractal analysis techniques can be valuable tools for the objective and quantitative characterization of abstract visual art. It is interesting to compare the fractal properties of popular art work with the same properties measured from natural objects. Is there a component shared between the artwork of Jackson Pollock and a beautiful mountain range? Further study using fractal techniques could help to quantify  beauty.






