Mandelbrot Genetics
by Jeffrey Ventrella www.Ventrella.com 
Read the Chapter: Evolving the Mandelbrot Set to Imitate Figurative Art in this book 

What the hell is this black thing? It looks like a squashed bug in the road. It's quite ugly in fact.
But wait a minute. Hold on. It's not ugly. Look more closely.
The images below are high magnifications of the boundary of this thing that we thought was just a squashed bug. They are colorized to make it easier to see the many levels of fractal selfsimilarity. Because of this endless variety of form, this thing  the Mandelbrot Set  has been called the most complex object in mathematics. So, depending on how you look at this thing, it can be quite beautiful. In fact, I might even say it's a little bit too beautiful. All this Psychedelic Baroque  it makes me want to do something subversive. 
As a young art student, I had a huge appetite for abstract expressionism and surrealism. I would gaze at the paintings of Francis Bacon, Robert Motherwell, and Arshile Gorky . I would marvel at their ability to mix beauty and ugliness in a way that forced my brain to see the world more clearly  more intensely. These artists would paint forms that lay somewhere between abstraction and representation. Gymnastics for the eyebrain system. 
Genetic Code
The Mandelbrot equation is z = z^{2} + c, where z and c are complex numbers and c is a location in the complex plane being tested. The function is applied many times, with the output value of z from each iteration being used as input for the next iteration. During iteration, if the value of z exceeds a magnitude of 2, then iteration halts and we declare that location in c as outside of the Mandelbrot Set (white). Otherwise, c lies inside of the Set (black). If the function were iterated only once at each c, the result would be a round shape, analogous to a single cell before subdividing into a multicelullar organism. Each time the function is iterated, the approximation of the Set becomes more refined, and the boundary reveals more bays and peninsulas. Fractal selfsimilarity increases. 
If instead of squaring the value of z, you cube it (z = z^{3} + c), the result is the form shown here. Many variations on the function have been explored, and the result is many kinds of "Mandelbrot Sets". These show how much variation there is in the world of iteration in the complex plane. Fractal explorers such as Clifford Pickover have created some pretty cool variations. 

Genetic Visual Language
As a part of the fractal journey that began in the mid 80's, I had generated a large series of images of a specific treatment of the Mandelbrot equation, resulting in a variety of organic, gestural forms. See my Mandeltweak web page for more examples. 
My total mutilation of the equation caused a mathematician who specialized in the Set to disregard my explorations as completely outside of any legitimate analysis from the standpoint of Complex Analysis. I took this as a sign that I was on the right track, and continued to mutilate the equation, exposing more and more parameters that enabled visual treatment, each parameter representing an adjective  a descriptor  of some visual concept. Little did I know that my irreverence to the mathematics would ultimately make me appreciate it more. 
Multidimensional Genetic Space
The relationships between these genetic variations is intriguing, similar to the way Richard Dawkins originally described the biomorphs in his Blind Watchmaker program. Dawkins' notion of a "Genetic Space" inspired me to create an art piece that was displayed at Galery Naga in Boston. To distinguish my Mandelbrot Art from what so many others were doing (Psychedelic Baroque) I decided to forget about color entirely, and render my images in GRAYSCALE. Its all about form  morphology. 
Consilience
In March '09, the three tweaks below, will be exhibited at the Pence Art Gallery in Davis, California, as a photographic trilogy called, "Fractal Self Portrait". Here is the accompanying statement to this work: Math is an unlikely medium for selfportraiture. But I have a special relationship with the Mandelbrot Set  regarded as the most complex object in mathematics. It is a magic piece of clay  pregnant with infinite form. I have tried to coerce it into imitating images of my face, using a genetic algorithm technique that I developed. The fractal details are readapted into facelike features. From a distance, one sees references to human heads. But there are obvious signs of a genetic signature that is not human. This is a methodology of genetic mimesis  inspired by evolutionary biology. 
In March of 2009 I began developing higherresolution images. These are shown below.
Click to see an enlarged image (warning  it might take a long time to download).

Find Way
In the fall of 2009, five images were exhibited in the show, Find Way at Point B Studio in Port Orford, Oregon. Here is a snapshot of the gallery space. 