When I display my art in galleries and art shows, I'm often asked for a more detailed description of how art can be made solely with mathematical formulas. Don't I mean that I've taken photographs or drawings and manipulated them somehow? "No," I patiently explain, "in all but a few rare images, my art is made entirely with mathematical formulas." How is this possible, they want to know? Based on most folks' experience with math, it's difficult to conceive of how such a process might occur.

The problem is that there isn't a similar art technique, or an ordinary mathematical application with which we can make a reasonable analogy. As with anything brand new, without analogies, or the ability to see a live demonstration, it's quite difficult to wrap our minds around something we've never experienced. Even if you could peer over my shoulder as I create an image, you still wouldn't have a clear conception of what was going on.

After time and much frustration, my impromptu explanations eventually evolved into one that begins with ideas and pictures that are familiar to everyone, even if the analogies aren't entirely accurate, and progresses into more technically-precise explanations along the way.

* Coloring in fractal-speak refers not to actual colors like red, blue and green, but to the general way in which the underlying fractal structure is made visible. When we need to speak of actual color, we'll use terms like gradient, palette, hue and tone.

I just want to note, at this point, that by altering the equations slightly, we could change the arc, size, direction, and angle of both the spiral pathway and the coloring shapes. We'll keep these for consistency throughout this demonstration, but a fractal artist always has the ability to alter those elements of every shape.

We still see the first iterated spiral of dots, but the five largest have now been replaced by five miniature spirals that resemble the original. The largest dot in the second iteration of spirals is the one that follows the shape of the original underlying structure. If this pattern were to continue for every dot in the first iteration, we would have two complete iterations of our soon-to-be-fractal spiral structure.

Up to this point, I constructed these images by hand, copying shapes and pasting them at reduced sizes along the underlying structure. Clearly this would be tedious and we would quickly reach the limit at which a human hand could draw the ever-diminishing shapes. But can you picture the concept that every dot in each iteration of our original underlying structure could be replaced by a smaller, self-similar shape? Because the shapes can be drawn by mathematical expression, and computers have a much more efficient and tireless processor than the human brain, the number of possible iterations is infinite — far beyond what we can actually see at once. But fractal structures can be "zoomed into", much like one can use a microscope to explore the minute details of biological or chemical structures.

If you're still with me, it's time to move from hand-drawn illustrations to a real, mathematically-generated fractal.

It's still difficult to see more than three actual levels of shape in an image of this size, but I can assure you that they're there.

But if you move leftward from the red dot, toward the blue side, you'll see the structure resolve into a very tightly-wound spiral. Together, the tightly-wound side and the gentle curl comprise our total underlying structure for this particular fractal. And now if you look at the smaller iterations emanating from each large dot, you'll see each contains both the tightly-wound and gently-curled paths.

The formula for this Julia set is:

Before we go any further, it might be good to remember that the entire underlying structure of this Julia set is determined by this formula. The elliptical shapes are determined by a second (coloring) algorithm. Given these two formulas to compute, a fractal generator such as Ultra Fractal performs the necessary calculations and renders the fractal onscreen.

And just as we colored our original spiral path with squares and abstract shapes (as above), we could choose different coloring formulas to apply to our Julia set instead of the elliptical shapes. But before we consider that, let's look briefly at the black and white gradient used in our illustrations thus far.

Here is the gradient we've used in our images thus far:

Note the sharp edges of the black and white sections. This corresponds to the sharp edges of the elliptical shapes in our fractal.

When indexed to our coloring algorithm and applied to our fractal, the ellipses take on a more three-dimensional shape. The varied tones in the background area give a sense of depth, as well. The fractal structure and coloring algorithms remain the same; the sculpting and lighting effects come from the manipulation of the gradient. Nifty, huh?!?

Notice how the blobs still follow the familiar tightly-wound spiral and gentle-curl pathways of the Julia in every iteration, at every size. Look closely at these two pathways now, because things are about to get more complex!

Since you know what to look for (or can refer back to the unadorned Julia sets above), see if you can still identify the largest tightly-wound spiral and gentle curl in each half. Although they're a little harder to identify in all this complexity, you should still be able to find the smaller repetitions of these two pathways as they comprise the larger iterations.

Coincidentally, the same coloring formula was used for this effect and the all-over spheres in the previous image. Just a change in two of the formula's parameters creates the different effects.

The previous five images illustrate four different coloring algorithms and five different shape and textural effects. If we save each as a separate layer in our image, we can begin to combine them together so that features of each contribute to a finished image.

By making the white background of the filament layer transparent, the background texture layer shows through, surrounding the clearly visible filaments.

You're probably thinking, "This is cool, but what about color?"

Hold onto your hat, here we go...

The gradient for the blob layer has three hues: a creamy white, deep fuschia, and green:

The one challenge to adding color to fractals, though, is that every repetition of a shape is indexed to the gradient in the same way, meaning that each has identical coloring. Consider each repetition of the ellipses and blobs in this image: they're identical. If I adjusted the gradient to include a different hue, every instance of that shape would be affected.

In this image, I've changed the gradients for each of the layers, but if you look carefully, I've also employed a technique* that alternates hues in the ellipse layer. While both of the largest ellipses are golden, if you follow the structural path in either direction, you'll see that the next ellipse is gray, then golden, then gray — continuing this alteration throughout.

* I teach this and many of the other techniques used in these images in my series of Ultra Fractal Courses.

The process of creating art from math is unlike any other visual art technique, and the learning curve is quite steep, but the magic and pleasure of exploring, manipulating, and creating fractal art is highly addictive!