Wait, “Math” and “Art?” If you’ve come this far, you may be thinking twice about these apparent opposites. Yes, it’s possible for the two to coexist and even complement each other.
This is a space to discuss various topics that don’t fit elsewhere regarding art, math, programmatic details, etc. I hope you find this interesting. Chances are, you arrived here through a search result. If so, please explore the rest of the site! I plan to add more to this page as my exploration continues in this area.
How Is This Art?
When creating fractal art, the computer does all the work. Really, You press a button and out comes an image. You may be asking, how is this art?
The art is in the painstaking preparation. For example, an exquisite bronze sculpture is created when a metal foundry pours molten metal into a cast. The metal cools, the cast is opened, and the sculpture is there! It isn’t that simple, though. An artist had to create the cast.
Computer-generated art requires an artist to set up the parameters. This requires careful exploration of the fractal space and fine adjustment of form and color. Like photography, framing and lighting are critical. It can take several hours (or days!) to create something that’s worth rendering to print.
To continue the bronze sulpture metaphor, multiple copies of the sculpture can be created simply by reusing the mold. This reduces the value of each copy, unless the mold is broken after the first successful cast. This creates a one-of-a-kind piece of work.
The same can be done with fractal art. If only one print is created from the work, and the parameters are never shared, then the work becomes unique. It’s possible to “break the mold” by securing or even deleting the parameters file and the rendered image file after printing. These become single-edition prints.
I have not had the courage to hit the delete button after finishing a work. The prints are one-of-a-kind, yet the original fractal provides opportunity for continued study. Unlike a mold, it is still a “living” work of art that can be explored further.
Motion
Creating moving art with fractals involves gradually changing one or more of the values that affects the underlying fractal formula, as well as the method used to render the fractal such as the approach to color the fractal form.
Possible Parameters
- X and Y coordinates in the complex plane
- The exponent e in the formula, z = ze + c
- When slicing through “multibrot” space, the slice angle and position
- Coloration using the red, green, and blue channels of each rendered pixel
There are two methods of rendering a moving fractal: rendering to video or real-time generation. With a modern graphics processing unit (GPU) that is built-in to today’s computers and devices, it is possible to generate motion in real-time on nearly any device. Higher complexity, such as varying the exponent in a trigonometric fractal, requires more capable devices such as the Layer Frame.
This is the next natural step for development of my fractal program, as all the components are in place. I have a little development to do…
Smoothing
Calculating a Mandelbrot-type fractal requires counting the iterations required for the formula to escape beyond a predefined value. This results in integer values, which when mapped produces a marked “banded” effect. A little calculus allows us to figure out how these bands are progressing overall and to smooth out the quantization effect of an iterating formula. This image shows the difference, the right half of the image being unenhanced.
Smoothing Formula
In most cases ln (2), 2 ∙ ln (2), and log (exponent) can be precalculated into constants to speed iterative calculations:
The iteration cycles must overshoot beyond the escape value a few times to collect enough samples for smoothing. This is a small price to pay for a more finished result.
Period-Mapping
Rendering negative exponents “breaks” traditional escape-time rendering of the inside of the set. Instead, periodic rendering must be utilized to render under negative exponents. Typically, this requires time-consuming sampling of many iterations for each point. However, Aleksandr Lyapunov developed a methodology to determine period-calculation without such sampling.
Applying Lyapunov’s work to the Multibrot universe, the calculation is fairly simple: take the natural logarithm of the average result and divide by the number of iterations run. No sampling, just apply a formula to the final figures. We can collect the sum over the iteration process without much extra CPU work.
Implementing this into a computer program, I realized that the formula can be simplified, I mean really simplified, with the goal of a satisfying visual result. The final periodic calculation used by this application resolves the average magnitude of a set of vectors. This provides a visual texture of the periodicity inherrent to negative exponents. My piece Sunset on Ice is an example of how complex this texture can become.
Simplified Periodic Formula
Since we are already calculating these squares as a matter of building the fractal, collecting the sums is a trivial addition (no pun intended). Adding one to the denominator avoids a runaway result to infinity for vector lengths close to zero. On the other hand, the inversion inhibits runaway values in the other direction. The overall magnitude of the result can be controlled by changing the constant numerator from one to another value.
These calculations are leveraged for optional effect-rendering on the inside of the set for positive exponents, and the outer area for negative exponents. Period-rendering can be applied at any time in my application through the Render menu. Also, variations of the effect can be selected through the Effect menu. The application can render both escape-time and period-mapping at the same time for interesting results. Try it with the trigonometric variants of the Mandelbrot fractal.
See the Wikipedia articles linked in the Resources section for more detail about smoothing and period-mapping.