\graphicspath{{artpsychology/}}
\section{Psychology of Fractal Art}

What is it about fractals that intrigues us?  Why are they aesthetically pleasing, and what can that say about our sense of aesthetics?  At this point, it will be helpful to call in the opinions of an expert on the subject, Sir Ernst Gombrich.  In his book, The Sense of Order, Gombrich explores what makes us perceive beauty in patterned art, and what kind of psychological effects patterns can evoke.  His writing makes no specific mention of fractals, but it is interesting to extend his ideas into the field of fractal art.

Gombrich makes an important point, that we cannot easily choose certain features of a figure as the reason behind its beauty.  The perception of beauty is largely dependent on who is viewing it, and in what context -- beauty is in the eye of the beholder.  We can, however, make some general observations about what kinds of things people find beautiful and explore the psychological reasons and ramifications.

One aesthetic quality that Gombrich describes is the concept of restlessness and repose.  A completely chaotic picture will tend to draw the eye in all directions at once, and leave the observer confused and uncertain.  On the other hand, a completely uniform and ordered design will leave the observer bored.  Humans appreciate order, and seek it in the environment around them.  Our attention is naturally drawn to places where a pattern is broken.  Fractals run dangerously close to being too chaotic to handle, but this danger is averted by self-similarity and patterns.  In this way, our eye is drawn to the repeating patterns to explore what makes each repetition different, such as rotation and scaling.

Next, Gombrich describes the sense of symmetry.  People tend to notice and explore symmetry, and art is often designed to take advantage of this.  Small departure from symmetry is unnoticed, but large departures are distracting.  Fractal images occasionally exhibit symmetry, and it is often beneficial to the appeal of the resulting image.  However, since fractals have mathematics at their foundations, they run dangerously close to becoming boring through perfect symmetry.  Fractal artists must be careful not to bore their audience with too much self-similarity and redundancy.

The metaphorical concept of movement in an image is another important psychological factor that Gombrich discusses.  Often, artists attempt to draw our attention and interest by designing figures that imply movement, such as pinwheels and spirals.  Fractals often contain spirals and whirls that naturally draw the eye along their path to the center.  This is enhanced because the spirals are often made up of self-similar copies of the main image, so the eye is first drawn by the similarity, then by the movement.  Spirals are often seen in the Mandelbrot and Julia sets, such as the center image of Figure \ref{fig:juliaTypes} on page \pageref{fig:juliaTypes}.

Another interesting aesthetic quality often found in fractal art is the feeling of depth.  By overlapping figures using positioning and shading techniques, the artist can strongly imply that one part of an image is behind another, invoking a three dimensional feel.  Examples of this can be found in the images of Charles Vassallo, one of which is shown in Figure \ref{fig:vassallo}.  His images contain strongly three dimensional objects that overlap and intersect each other, giving an overpowering feeling of depth.

\begin{figure}[htbp]
\begin{center}
\includegraphics{vassallo.png}
\end{center}
\caption{One of Charles Vassallo's images of Markus-Lyapunov fractals.}
\label{fig:vassallo}
\end{figure}

A very important factor contributing to aesthetic appeal is the use of color.  Gombrich argues that, without color, the effects of an image are severely limited.  Color is a key part of the fractal rendering process, manipulated carefully by every fractal artist.  Artists must ensure that the use of color is not chaotic and confusing as in some of our pictures of the Mandelbrot and Julia sets with a rainbow palette.  Color can be carefully manipulated to provide depth and meaning to a piece.  It is also a very important factor in separating ground from figure.  Fractal artists manipulate their color palettes to emphasize the parts of the image they find most interesting, while muting the rest, so that the important object stands out.  Without this, the viewer would have little to focus on, and the image would seem chaotic and meaningless.

This leads to the final important aesthetic tool used by fractal artists: object recognition.  Gombrich explains that humans naturally try to find objects they relate to and understand, and this is evident in the process of many fractal artists.  They often describe the process of searching through the Mandelbrot set, identifying recognizable constructs such as seahorses and spirals.  Fractal artists often try to find recognizable features and modify their rendering method to make an object more familiar.  They will occasionally go so far as to give suggestive names to their works, such as Janet Parke's ``Iron Gate''.  It seems as if fractal enthusiasts find some pleasure in finding a recognizable object in the purely mathematical world of fractals.

Fractal artists are careful to consider these aesthetic properties when they create their art.  These are the kinds of qualities that separate the work of a true artist from an amateur fractal hunter.  Often, an amateur's work will have very few of the above qualities and may even accidentally break some of the ``rules'' and create something that is actively displeasing or disconcerting.  A fractal artist is careful to consider the effects of his work on its audience.

\subsection{Vassallo's art}

Looking back at Figure \ref{fig:vassallo}, what can we say about its artistic characteristics?  Charles Vassallo's work has been featured in an exhibition in Lannion, France, and has been submitted to several art competitions.  His images seem to depart from the purely mathematical and encompass more of what we might call traditional art.  What might Gombrich have to say about this example image?

This image presents a smooth, unbroken object that instantly stands out to the viewer.  The eye is drawn across its convoluted contours and to the intersection points, because this is where the continuity of the object is broken.  Our eyes are naturally drawn to examine discontinuity, so we will tend to take the long, smooth areas of the object for granted, and examine the intersections and overlaps more closely.  In this way, Vassallo carefully manipulates our discontinuity sense to draw us to explore the three-dimensional qualities of the object.  There is just enough continuity to keep the image from feeling restless, and there is a balancing discontinuity to keep our eyes moving.

The mathematical idea of symmetry does not apply to this figure.  Nevertheless, we can still apply some of the opinions of Gombrich to this image.  He suggests that symmetry can have both positive and negative sides.  While Vassallo's image does not exhibit symmetry, it also does not disrupt our sense of symmetry.  For example, a picture of a roughly symmetric object might distract our attention if it exhibits key asymmetry that is bothersome to our sense of order.  Vassallo's object is abstract enough that the mind doesn't even try to find symmetry.

When speaking of movement in a piece of decorative art, Gombrich used the examples of waves, pinwheels, and spirals as objects which imply movement to us.  In the background of this example image, we see a wavy purple surface reminiscent of the ocean, giving a strong suggestion of a viscous wavy liquid.  The foreground object also gives an impression of twisting and contorting on itself, because of its convoluted structure and strong shading.  It is likely that the artist was well aware of the sense of movement that he was creating, and that he intended to hold the viewer's fascination for as long as possible.

This image, among all of Vassallo's available works, was selected based on its extraordinary sense of depth.  The artist has carefully shaded the central object to give the impression of a light source, with shadows at overlapping points to suggest that the it folds behind itself.  The object itself is abstract, but its depiction of depth and solidity is so concrete that the viewer can easily imagine himself holding it.  Our visual system would easily notice if, for example, the light spots could not have resulted from a single light source, and this would likely have given us a feeling of unease or confusion.  The artist clearly had to carefully develop the depth of the image, so that the light areas and shadows do not conflict with eachother.  

To achieve the effects of movement and depth, Vassallo has made careful use of color.  First, he has separated figure from ground by choosing two distinct colors, purple and yellow.  The viewer's eyes are instantly drawn to the central yellow object, which is accentuated by the visual cue of the shadow cast on the purple background.  Further careful color use creates the sense of depth and solidity in the object.  It is no mistake that only two major colors are used in the image.  Any more might cause the image to lose its continuity, making it feel more restless.

Finally, how does object recognition apply?  It is easy to see that Vassallo is trying to imply some kind of solid object with his image.  But this object is completely fictitious, and might possibly play games with space, like the work of M. C. Escher.  In a way, it is somewhat distracting that the image depicts an object that the viewer cannot identify.  However, the image does not seem to claim that it depicts reality at all; rather, it tries to suggest some kind of mathematical fantasy-world.  So, the fact that the viewer cannot identify with the object is not in any way distracting.  It keeps the viewer interested, but not disturbed.

Vassallo's work seems to share many qualities with established works of art.  In analyzing the qualities of this image, it is clear that the artist was very much aware of how his image would affect its audience.  He carefully expressed an idea through his work, and it tends to create emotions of intrigue and fascination.  It is the authors' opinion that this image is indeed a work of art.

\subsection{The John IFS}

Recall the fractal created by one of the authors shown in Figure \ref{fig:ifsjohn} on page \pageref{fig:ifsjohn}.  This was created primarily to explore the mechanics of Iterated Function Systems and as a whimsical toy picture.  A major focus of this project is to try to decide whether fractal images can be considered works of art.  We've shown by example and argument that the answer is not a definite ``no'', but perhaps an analysis of the John fractal can show that not \emph{all} fractals are works of art.

The John fractal is a purely mathematically defined object, completely described by 91 numbers.  It is no surprise that it exhibits a sense of repose so strong as to be boring.  People have plenty of experience reading words, even sideways and mirror-imaged.  After the initial reading of the image, there is almost nothing left for the eye to explore; the image just sits there.  There are no breaks in the pattern to draw our interest, because of its nature as an IFS.

The image exhibits a form of symmetry in its self-similarity.  Every occurrence of the name John is made up of smaller versions of the whole image.  This is intriguing when the initial concept of infinity sets in, but the feeling of fascination doesn't last for very long.  Quickly, the mind becomes bored, because the self-similarity is too extreme.  Gombrich points out that symmetry should be tempered with a certain amount of deviation to avoid boredom.  For a second time, we have found a reason why the John fractal is likely to bore its audience.

The fractal also fails completely in the area of movement and action.  The fractal is completely devoid of any kind of movement.  There are no suggestive shapes, not even a spiral down to infinity that is often a product of the definition of an IFS.  This is missing in part because only three levels of self-similarity are distinguishable at the level of detail rendered.

Depth is yet another area in which the John fractal is lacking.  Especially with its haphazard coloring and white background, there is nothing to suggest that the fractal is in any way three dimensional, such as overlapping, shadowing, or intersection.

The coloring in this fractal is also a product of its minimalistic mathematical definition.  The color of each pixel is chosen by the affine transformation used during each iteration.  It is no surprise that this creates a color palette that adds nothing to a feeling of movement and depth in the image.  Color is used solely to separate figure from ground, and no care is taken to avoid the appearance of randomly chosen colors.

About the only thing the John fractal has on its side is object recognition.  The name ``John'' is instantly readable, and the concept of infinity is fairly clear after a few seconds of consideration.  The author was careful to position each word so that it is easy to read, avoiding upside down and mirrored text when possible.  The letter ``O'' was carefully crafted to be symmetric.  These choices were made because inconsistencies in the direction of the text in earlier versions tended to draw attention to certain areas, distracting the viewer.

It is fairly clear that the John fractal is not a work of art.  It can best be considered a mathematical joke, worth a few seconds of interest.  Unlike the previous example, it does not draw any emotion, and it almost actively bores its audience.  There is nothing to keep the viewer interested in the image beyond the first glance.

\subsection{Janet Parke's ``Then I Saw a New Heaven''}

For several years, Fractalus.com hosted a fractal art contest.  The 1999 contest drew entries from over 130 fractal artists and selected winners in 13 different categories, such as Emotion, Best Color, Organic, and Mechanical, among others.  Janet Parke  won several awards for two of her three submissions.  We will examine her entry, ``Then I Saw a New Heaven'', shown in Figure \ref{fig:heaven}, the first place winner of the Quote Interpretation section of the contest.  The image is an interpretation of Revelation 21:1,4 from the Bible:

\begin{quote}
Then I saw a new heaven and a new earth; for the first heaven
and the first earth had passed away... and death shall be no more,
neither shall there be mourning nor crying nor pain any more, for
the former things have passed away.
\end{quote}

\begin{figure}[htbp]
\begin{center}
\includegraphics{heaven}
\end{center}
\caption{Janet Parke's ``Then I Saw a New Heaven''}
\label{fig:heaven}
\end{figure}

Her image depicts heaven as a cloudy white fractal figure in the center, surrounded by serenely colored skies.  This is framed by a silvery substance reminiscent of mercury, which appears to be opening up to expose Heaven.  Another interpretation might be made, but the title and Bible passage tend to guide the mind into perceiving the image in a specific way.

The colors and textures clearly create an image that is not as boringly static as the John fractal.  There is a degree of restlessness, created by the rippling structure of the metallic liquid.  However, this is not so powerful as to distract the eye from the image of Heaven.  Instead, it gives the feeling that the liquid is flowing away to expose Heaven in the center.

Symmetry does not apply to the outer frame, but it can be found in the white clouds in Heaven.  The clouds spiral into the center of the image, providing self-similarity.  It appears that the similarity is mathematically perfect, and that this is a fractal object processed and colored to look like clouds.  Though the similarity has no breaks, it does not fall into the trap of being boring, because it does not try to be the main focus of attention in the image.

The image, with its suggestive title, invites a feeling of movement.  It seems to imply that the viewer is swimming through some kind of obstruction.  With the context provided by the title and Bible passage, the suggestion is that viewer is moving forward toward Heaven, leaving behind the oily substance of death, mourning, and pain.  This is accentuated by the carefully manipulated structure of the foreground matter, which is made to look like a liquid that is sliding away from the center of the image.

The layering of the picture also gives a sense of depth.  It is clear that the dark, negative foreground liquid is close to the viewer and that Heaven is further away but close enough to reach for.  The artist made careful use of transparency so that Heaven is viewable between all of the gaps in the metallic liquid, making it seem as if the darkness is obscuring Heaven.  

Color is used in this image primarily to express emotion.  The negative emotions of death and pain are depicted as a filthy oily substance obscuring the view.  Heaven, on the other hand, is represented using soft, tranquil colors, giving a sense of peace and happiness.  Color is also used to define objects that the viewer can recognize, such as the fractal structure in the center of heaven that is colored to look like a cloud.

It is clear that Janet Parke kept in mind how her audience would react to her image.  While the ``JOHN'' image is merely a fractal rendering, ``Then I Saw A New Heaven'' is a carefully created work of art, with fractals as the paintbrush, and layering techniques as the canvas.

\subsection{Dots}

In an attempt to create some original art work, Steve Conte developed the image ``Dots'', shown in Figure \ref{fig:dots}, using Ultra Fractal.  The authors have little experience creating art, and do not consider themselves artists.  Nevertheless, ``Dots'' is somewhat interesting to look at.  In order to fairly assess the artistic merit of this image, it will now be analyzed by John Waymouth.  He has been given no knowledge of what process was used to create the fractal so that his opinions are not swayed in any way.

\begin{figure}[htbp]
\begin{center}
\includegraphics{steve3}
\end{center}
\caption{``Dots'', by Steve Conte}
\label{fig:dots}
\end{figure}

At first glimpse, this image seems as if it suffers a similar problem as the John fractal: its sense of order and repose is dangerously close to boring.  However, one's eye is soon drawn to the dot field in the lower right, and the eye starts to try to make some order out of the pattern.  Phantom arcs seem to connect the dots in rivalling orientations, and the eye is also drawn to explore the clusters of dots.  But after this initial surprise, the image settles down again, and patterns tend not to have much more interesting information to divulge.

Though it may not be obvious at first glance, this image has reflective symmetry along an axis approximately $20^\circ$ counterclockwise from vertical, which is more noticeable if the viewer tilts his head to match it.  Any benefit that might be gained from symmetry is lost because it is not readily apparent, due to its non-standard orientation.  On the other hand, this symmetry is mathematically precise, and Gombrich warns against this kind of symmetry because it is boring.  Perhaps the mind unconsciously notices the symmetry, and this contributes to the overall emotional feeling created by the image, but I did not consciously notice it until I specifically looked for it.  It seems likely that this symmetry is an accident of the rendering technique used, and not a decision of the artist.

This image has a definite sense of depth, created by the use of dots of varying size.  As the viewer's eyes follow the movement of the nearest set of dots, he might feel the need to move into the picture and follow the path.  The nearest dots obscure the view, and appear hazy and indistinct, as if the eye is unable to focus on an object so close.  

The coloring seems to be as accidental as the symmetry.  Red is used for the repeated visual element of the dots, and a rather plain blue is used for the background.  In the works above by Parke and Vassallo, the background always had some interesting detail, such as gradients or waves.  In ``Dots'', the background is monochrome, and this lack of detail seems to paradoxically attract attention.  When the eye moves to examine the sea of blue more closely, it is not rewarded with any new detail, and this detracts from the emotional experience and leaves a feeling of shallowness.  Perhaps black or white would have been better choices, because they are colors more traditionally used as plain, undecorated backgrounds.

``Dots'' has no appeal for the object recognition sense that is part of the human visual system.  The only structural object in the image is the dot, repeated hundreds of times at varying distances.  Its uniformly hazy, circular appearance has no detail to interest the eye, and seems to exist only to define some three-dimensional starfield.  There are no objects to engage our interest, except those created by the juxtaposition of groups of dots.  The only interesting patterns are in the arcs described above, and the five central cables that extend off to the horizon.  Since the termination of these objects is obscured by one of the dots, the eye is not rewarded by its natural tendency to follow the ``dotted lines'' to their conclusion.

Steve Conte's ``Dots'' and the John fractal seem to be lacking some underlying quality that is present in the previous examples by Vassallo and Parke.  In both of these images, there is visual information that keeps the viewer interested beyond the first cursory inspection.  It is possible to catch attention with simple fractal patterns, but it seems to take a true artist to hold our interest and make us want to explore an image more fully.  It is clear that fractal artists must carefully consider how their work will be perceived.  They must make careful use of the psychological impact of patterns, symmetry, movement, depth, color, and object recognition to stimulate the interest of their audience.

This is why ``Dots'' and the John fractal are not nearly as artistically interesting as the works by Parke and Vassallo.  Since we are not talented and experienced artists, we did not know how to design our images to benefit from the psychological impact that various visual elements have on the audience.  When examined as works of art, our fractal images appear just as amateur as any of our artistic undertakings in other media, because we are not accomplished artists.  
