"f[z]=arcsech[sec[z]/5]": {x, -6.5, 6.5}, {y, -1, 1}

Fractals are some of the most beautiful and interesting shapes that the study of mathematics has produced. Technically, they are self-similar geometric sets with an infinitely fine and irregular structure at arbitrarily small scales. The self-similarity can be exact, as in the case of the Cantor Set, or approximate, as with the Mandelbrot Set or a coastline. With exact self-similarity, a precise rule is followed so that any chunk taken from the set or shape is exactly the same as any other sized chunk. Approximate self-similarity obeys a more general rule of growth, and has more room for variation. A good example of this is the way a tree grows, the branching at the finest level is self-similar to the branching at the largest scale.

Some fractals exhibit the unusual trait of having finite area and infinite circumference; they might be represented by a function which is everywhere continuous and nowhere differentiable. These are a new type of shape described at the turn of the 20th century as being “pathological” or “monsters” by mathematicians first seeing objects such as Hilbert's space filling curves or the Koch Snowflake.

To the layman, they are whimsical shapes which seem to repeat themselves in such a way that no matter how far one zooms out or in, the essential structure remains the same.

We see fractal-like designs in nature all around us, in cloud formations, lightning bolts, even in stalks of broccoli and cauliflower. There is as much art to fractals as math, and they are one of the few highly complex mathematical procedures which can be translated into an experience someone with no math background can thoroughly enjoy and delight at.

The purpose of this research project was three-fold:

1. Study some of the mathematics behind fractals.
2. Create art objects as a reaction to the mathematics learning process.
3. Discuss the connections between mathematics and art in general, and reflect on the specific ones that surface in this learning-creative exploration.

This process included forays into computer science to learn some of the programming required to generate these types of images, a study of the principles of color and design for the art aspect, and discussion of the validity of math objects as fine art. Falconer's text, "Geometry of Fractal Sets" was used as a basis for the mathematics and many pre-eminent authors were read and discussed over the Fractal Crew's meetings.

The links on the left take you to information on the topic and art pieces inspired by that particular mathematical idea. Many thanks to Rockhurst University, Dr. Guadarrama, the Dean's Fellowship and the James and Elizabeth Monahan Fellowship for supporting this project.