History

Are the fractal images shown in this website, or many others scattered throughout the world-wide-web, art? That is the question to be addressed in this discussion.

Chaos and Fractals

Some basic science and mathematical ideas need to be presented in order to appreciate and understand what fractals are and how they can be created. The notion of “chaos” is a starting point. Weather, earthquakes, leaves falling from a tree into a stream are examples of chaotic systems. They are dynamic. For weather the spin of the earth, surface and air temperatures, the chemical make of the atmosphere interacts to create incredibly complex patterns. The earth’s molten core is in a spinning pot and it moves around putting pressure on the surface crusts or tectonic plates. As these plates rub against one another the cause earth quakes and volcanic eruptions. These phenomena cause cracks, valleys, mountains, oceans, tidal waves and a host structures and features that are records of the chaotic, dynamic activity. These records or tracks can be described by fractal geometry.

I like the following wording from “Fractals: the patterns of chaos” by John Briggs.

“So a fractal is the fracture left by the jarring of an earth quake or the winding coastline printed with the turbulence of the ocean and erosion; it’s the branching structure of the fern which traces the process of its growth; the scrambled edges of ice as it freezes; the spacing of stars in the night sky; the clouds and plumes of pollution spreading out from a power plant. When a chaotic thunderstorm self-organizes into a tornado, it leaves behind it a fractal shape of its destruction.” (p22, Fractals, J. Briggs)

Fractal geometry was invented by Benoit Mandelbrot, an IBM researcher, in the 1960s-1970s. He used the term “fractal” to suggest “fractured”. Fractal geometry was created to describe broken, wrinkled, and uneven shapes. Fractal geometry allows the visual modeling of coastlines and mountains and many structures we see in nature such as soil erosion and seismic patterns.

Fractals have two fundamental properties: self similarity and non integer dimension. Self-similarity means that we can magnify or reduce scale and at every step one can see essentially the same shape. Essentially same shape means one of three things: exact self-similarity, almost self-similar or statistically self-similar. For example, the shoreline created by the ocean is similar to the shoreline created by one of the great lakes, or smaller lakes. The Mississippi river is similar to the Detroit river or a stream with respect to flow, the way the river or stream forms its course by cutting valleys over time. Snowflakes are similar based on 6-pointed geometry. Mountains are digitally created for movies as are flowing rivers.

As for dimension, a line has dimension of 1. A plane is 2-dimenaional, x-y coordinates but zero thickness, 3-dimensional objects are just about everything we can hold and touch. Fractals can have dimensions such as 2.3 or 1.7, numbers between integers. We can’t really comprehend what this means except to say the fractal dimension is a mathematical dimension used in the formulas that create fractal visual models that are the images seen in the fractal art collections.

The Mandlebrot Set, which was derived and created by Mandlebrot, has a dimension between 2 and 3. It is shown and labelled in the image of the web-page that brought you here. If you zoom in on any point on the boundary of the set it has a jagged outline. The set obviously has a fixed area, you can see that by looking. However, its boundary is infinite. No matter where you zoom to on the boundary you get a similar jagged line.

This is not the place to go into all the equations that can be used to create the visual models of fractals – the digital fractal images present in the collections. Suffice it to say, one can create extensive libraries of fractal geometry equations for the visual modeling of the fractals. It is these kinds of equations that the artist can use to create patterns and the associated coloring pallets.

Fractal Art

Fractals have been described as “never ending patterns”; like snowflakes, lighting strikes, galaxies and waves on a beach. As an artist and engineer I find such patterns fascinating. While nature’s patterns have always inspired artists, the nature of fractals is just beginning to capture the imagination of artists. The engineering part of me loves how computers can create incredibly beautiful fractal patterns from a mathematical score. Like a symphony, fractal patterns can be simple or complex, peaceful or tumultuous, whimsical or serious.

Digital fractal art creates patterns using a variety of mathematical equations combined in a computer program which forms images by calculating patterns over and over in an ongoing feedback loop. Coloring algorithms are used to add colors to these emerging patterns. The artist can stop the computer’s image creation process at any time and explore the layers of detail to select a portion of interest for further work. The selected image can then be refined using Photoshop or other digital graphic tools.

The fractal artist composes the symphony; a collection of equations in a program. The equations are the paint brushes and the color palette. The notes are colored pixels on the computer screen which displays the resulting digital image. Like an artist applying paint on a canvas, randomness can be included in the equations. As a painter controls the degree, velocity, and area of placement of applied paint the equations can be programmed to control the degree and nature of randomness from highly controlled to purely random. In this way every newly created “symphony” is a unique creation. While the original Mona Lisa can’t be replicated, her image can be duplicated. The score, or program, of each fractal symphony can be saved and replayed, much as a symphony can be performed by countless orchestras. And as each conductor with each orchestra will interpret the composers original score or perhaps even modify it to create different arrangements, so to the original score of a digital fractal image can be modified.

More Information About Fractal Art

Fractal Foundation: Mission: Use the beauty of fractals to inspire interest in science, math and art.

Fractal Software: This list of fractal creation and exploration software is available on the Fractal Foundation website.

References

1. Briggs, J., Fractals: The patterns of chaos: A new aesthetic of art, science, and nature. 1992: Simon and Schuster.

2. Garousi, M., the Postmodern beauty of fractals. Leonardo, 2012. 45(1): p. 26-32.

3. Mandelbrot, B.B., Fractals and an Art for the Sake of Science. The visual mind: art and mathematics, 1993: p. 11-14.

4. Spehar, B., et al., Universal aesthetic of fractals. Computers & Graphics, 2003. 27(5): p. 813-820.