\section{Are Fractals Art?}

Before we explore different kinds of fractal art, it is important to consider this question: can fractals, in fact, be considered a form of art?  Certainly one can see, by flipping through the diagrams later in this report, that there are plenty of examples of intriguing and beautiful fractals.  But the question remains, can we place our work in the same field as da Vinci and Michaelangelo?  Can the process of manipulating mathematical formulas be likened to a process of artistic creation?

At the outset, some people will argue that the answer is no.  They claim that, because fractals have mathematics as their basis, there is no chance for a person to introduce meaning and intrigue.  Mathematics is, after all, a dry field of unwavering truths and equations; numbers don't express emotion.  Others might point out that fractal images exist only as the creation of a computer.  Computers follow specific instructions and cannot think for themselves or experience emotion.  Or perhaps opponents will point out the fact that the fractal image was ``always there'', just waiting for the mathematician to stumble across it.  Finally, perhaps conceding that there are infinite possible images to be found in a given fractal, opponents have argued that the ``artist'' is merely punching in random numbers to create his images.

The purpose of this project is to show, by example and by discussion, that fractal imagery is an art form in and of itself.  The number of possible images in each fractal is infinite, and not every one of these is even remotely aesthetically pleasing.  But some fractal images are strikingly beautiful.  They have infinite detail that can always yield new shapes at closer inspection.  Their self-similarity tantalizes our natural need to make associations between concepts in our minds.  They can be intriguing, and undeniably beautiful\ldots but are they art?

What \emph{is} art, anyway?  Authors have written long philosophical volumes attempting to tackle that question.  Indeed, the question is beyond the scope of this project.  We will have to settle with a working definition, and use this to give the reader a sense that there is art in fractals.  

We know that certain works by the great artists strike us as beautiful.  For some reason, a picture will intrigue or fascinate us.  Other works communicate emotions to us, or try to make a point.  Some works, like the drawings of M. C. Escher, try to catch our interest with a visual riddle.  All of these qualities can also be found in fractal art.

But what about the idea of artistic process?  Fractals are static images, generated predictably from a specific set of input numbers.  How do we get past the arguments above?  To see how, we must examine the procedure involved in creating fractal images.  There is a process of trial and error, of experimentation, of making small refinements to make a more interesting picture.  The mathematics may stay rigid, but the pictures don't create themselves.  A fractal image is only a visual realization of the underlying mathematical process.  It is by varying the method of visualization that the artist can express himself.

Is photography a form of art?  After all, one is only reproducing what is already around us.  How can we call photography art, if the photographer is merely copying what we can already see?  But photography is already a well accepted form of art.  The photographs of Ansel Adams are very popular, and many art institutes have entire departments devoted to photography.  Photography is art, but why?

Pictures don't take themselves, and they don't develop themselves.  A photographer chooses what to shoot, what lighting to use, what shutter speed, what kind of film, where to focus, and so on.  Then he chooses how long to expose the film, what kind of developing techniques to use, how long to develop the film, how large to make the print, and many other factors, some of which are unique to each photographer.  Here lies the heart of the matter: photography is a form of art because the photographer uses his own taste and aesthetic sense in choosing how to make his picture.  We may not be able to define beauty and art, but we can sense it in our creations.  We can learn from experience how each factor affects the end result, and therefore, we can express ourselves through our art.

Fractal art is very similar.  A fractal artist must choose what kind of fractal to use, where to zoom in, what iteration algorithm to use, what kind of coloring to use, how large  an image to use, how much detail to capture, and other features.  Some fractal artists even take their raw fractal images and edit them in a graphics editor, which allows for further expression.  A fractal artist is not pressing random buttons and seeing what comes out, he is experimenting with many kinds of variation, and deciding if he likes the end result.  In this way, fractal art is much like photography, and, indeed, like any other kind of art.

Another reason why fractals hold artistic appeal is that they are closely related to the manner in which the world around us works.  As we will discuss later in this paper, fractal geometry can describe many natural formations much more accurately than traditional geometry, which is based on straight lines and geometric shapes.  Fractal functions have been crafted to look like land masses and mountain ranges, and such images have been successfully used in movies.

Fractal images can strike us on an instinctual level because they resemble the way things work in the real world.  As a child, John remembers watching with fascination a short segment about fractals in a children's show.  The imaginary camera kept plumbing the depths of the Mandelbrot set (described in the next section), demonstrating the concept of self-similarity.  What was once a tiny detail in the entire shape turns out be a copy of the whole shape in miniature.  And the tiny details in that shape, in turn, are copies of the whole shape again.  No matter where the camera explored, more detail could be found, always with the same recurring themes.  

Now imagine looking at a greatly detailed picture of a cloud.  Under a microscope, the picture would demonstrate that, what looked like a small offshoot of the main cloud is, in fact, reminiscent of a cloud itself.  And a small offshoot of this sub-cloud looks like a cloud too.  Now consider a head of cauliflower.  Look at the whole thing, and then break off a piece of it.  Examine this piece, and notice that it remarkably resembles a whole cauliflower plant.  Break off one of these pieces, and it, too, resembles its own plant.  The similarity stops after about three levels, but it is still intriguing.  Fractals exhibit this self-similarity down to infinity, while nature exhibits it on a limited, but still significant, scale.

Coloring techniques also help add to the beauty of a fractal.  High contrast can highlight certain details of a fractal and can help self-similarity to stand out.  We can mimic the vibrancy of nature in our fractals, and we can use this as a tool to heighten a fractal's similarity to a natural form.  Smooth gradients help guide the eye from one part of a fractal to another.  Bright, wildly changing colors catch the eye and encourage exploration.

Fractals provide a new medium for artistic expression.  Fractal artists learn the skills necessary to express themselves through fractals, and, in this way, they have opened a new field of art.  While we, the authors, are not artists, we hope to demonstrate our own explorations in this growing field and give the reader an idea of the nature of this new art form.
