\section{Introduction}
The field of mathematics has seen many recent discoveries related to fractals.  Fractals hold many secrets that mathematicians are only beginning to unlock.  From the intricate and colorful Mandelbrot set to the infinitely repeating pattern of the Koch curve, fractals hold the interest of many.  Recent discoveries have found fractals and similar formations in many areas of research such as physics and geometry.  Other links can be found between fractals and natural formations such as mountain ranges.  Fractals can even be found in works of art, where a painting has more detail visible when you examine its edges arbitrarily closely.

This Interactive Qualifying Project (IQP) focuses on the artistic possibilities of fractals.  It is the goal of this IQP to explore the applications of the growing field of fractals to the arts.  We will give a general introduction to fractals, and then show how these ideas relate to various areas of art.  In this way, we intend to demonstrate that, along with having intriguing scientific ramifications, fractals have a very important humanistic side.  This, we feel, relates science and sociology, which is the purpose of an IQP.

The authors hope to broaden their own knowledge of fractals as well.  This project provides two interesting opportunities: we not only will explore what mathematically defines a fractal and what mathematical properties they have, but we will also see how fractal images can be considered works of art.

Science thrives on the benefits it gives to average people.  There would be no purpose to scientific discovery if it did not bring improvement to the lives of many people.  Medicine is an obvious example: medical discoveries lead directly to improving the health of patients.  Fractals can also provide a benefit in that even purely mathematical fractal images such as the Mandelbrot and Julia sets are strikingly beautiful.

The authors hope to broaden the impact of their existing scientific knowledge by seeing how it can positively affect peoples' lives.  Instead of studying only the pure mathematics of fractals, we will forge a link between mathematics and beauty.  It is our hope that we can gain some experience with what beauty itself is.  In this way, our future scientific efforts will be improved because we will be more attuned to how our efforts will be viewed by an average person.  And as was outlined above, this is arguably a very important consideration when exploring science.

One can find many examples of beauty in fractals.  It would not be effective to try to cover all of the many types of fractals at once, and it is beyond the scope of an IQP.  However, we feel that we can gain some of the insight described above by focusing on a few links between fractals and art.  By exploring some examples, we hope to show how fractals and art are deeply related as a whole.

\subsection{Fractals}

The term ``fractal'' encompasses a somewhat loosely defined type of mathematical object.  Traditionally, fractals are defined as objects exhibiting self-similarity; that is, objects containing parts that look like the whole when magnified.  Objects like the Koch Curve and Iterated Function Systems, which will be explored later, are self-similar objects.  When using this definition of ``fractal'', it is implied that these objects must have new detail on every scale of magnification.  In fact, fractals are commonly described using just this last criterion: that they have fine detail of a similar structure at every scale of magnification.  This is the definition used for this IQP, which allows us to explore such objects as the Mandelbrot and Julia sets, which do not exhibit true self-similarity.  

The fine detail common to fractal images is what helps to make them beautiful and fascinating to behold.  Fractal images tend to be wildly chaotic, with new and interesting formations that draw the eye.  They usually have vivid color and often exhibit self-similarity, or at least some general form of self-mimicry, at many levels.  The eye is often drawn to explore a tiny detail, which, on closer inspection, is very similar to the image as a whole.  This plays on the natural human instinct to correlate and explore similar visual stimuli.

It is more interesting, for artistic purposes, to concentrate on the representation of fractals in a computer graphics environment.  Fractals themselves are infinitely detailed mathematical constructs, so it is not possible for humans to visually and aesthetically experience a fractal object.  It is the rendering process that brings some sense of order out of chaos: it gives us a way to visualize a fractal.  It is important to understand that the images produced by processes described in this IQP are not the fractals themselves; they are just a graphical approximation of the fractal object.

As we will demonstrate, the rendering process plays just as important a role in creating an aesthetically pleasing image as the fractal formula itself.  To represent a fractal graphically, it is necessary to make many choices, such as color, size of image, parameters, rendering effects, and many other possibilities.  Choosing a good rendering method can produce an image of startling depth and beauty, while choosing a poor rendering method can create an image that seems like a random collection of differently colored dots.  This project will demonstrate several popular fractal rendering methods and explore their aesthetic appeal.
