Three Faces of Pi

Fractals aren’t just shapes; fractals can be numbers as well. A fractal sequence is one that, in some sense, contains infinite copies of itself. One example of a fractal sequence is the signature of an irrational number (a number that can’t be expressed as a fraction, like π or the square root of 2). To create a fractal signature sequence for a number x, create values of y, where y = i + jx and i and j are both integers. Create y values for i and j from 1 to infinity and order them from smallest to largest. If x is irrational, then every y value will be different and the sequence of the i’s and the sequence of the j’s are both fractal sequences.

Infinity is really big, so instead, let i and j both vary from 0 to 32. Then, draw a line from (0, 0) to (32, 32), going through each point in order of the y values. Different irrational numbers will create different graphs. Even though each graph is one continuous line, the angles of the zigs and zags change, giving the appearance of rectangular blocks of different shades of gray.

The above image is composed of three separate graphs, aligned so that the entire image is one continuous line. The graphs represent three different irrational numbers which are based on π: π/2 (~ 1.5708), the natural logarithm of π (~1.145), and the square root of π (~1.772).

Standards

I had an unfortunate laundry experience this morning (turns out that “permanent press” is more wishful thinking than truth in advertising). I decided to entertain myself while ironing by listening to Eric Clapton tear through some blues standards (if you have to iron, this definitely helps). This got me wondering: Are there any “standards” in fractal art? That is, if fractal art is likened to blues or jazz or popular music, are there any images that might be the equivalent of “I’m Your Hoochie Coochie Man,” or “Take Five,” or “Night and Day”?

It seems to me that a fractal art standard would be an image intimately familiar to anyone intimately familiar with fractal art and vaguely familiar to those with less connection to the art. It would include a relatively simple formula, to increase its popularity and ease of calculation with various coloring schemes. Coloring formulas might be like genres in the musical analogy. (Here, I’m thinking in the Ultra Fractal schema, but such an image must not be tied to a particular platform.) The zoom and parameters should be such that the image is generally recognizable, but they could be varied as part of the artist’s rendition of that image.

Immediately, the overall Mandelbrot set leaps to mind. Zooms might be of the West Midget, the Seahorse and Elephant Valleys, and the head. Other candidates:

• Newton’s Method for third and fourth roots of 1
• Hilbert curve
• Koch curve/snowflake
• Sierpinski gasket
• Barnsley’s Fern

The Joy of Apathy

I recently realized that I just don’t care anymore. I don’t care if fractal art is Real Art (tm), if algorithmic art is fractal art, or if digital art is fine art. I don’t care if digital prints are multiple originals, reproductions, giclees, photographs, limited editions, or edited limitations. I don’t care if my art is any good, if your art is any good, if any art is any good. I don’t care if real mathematicians think I’m a fake for playing with pictures and not having a degree in math or if real artists think I’m a fake for playing with numbers and not having a degree in art. I don’t care if the “fractal elite” are personally leading a conspiracy to take over the world (although I think it’s cool that Damien Kenobi has apparently perfected the Fractal Jedi Mind Trick).

That’s not to say that I don’t care about anything, or even that I don’t care about my art. It’s just that after 20+ years playing with fractals, chaos, complex analysis, number theory, numerical analysis, image processing, computer programming, geometry, algebra, and calculus, I know what I like, I know what I like to do, and I’m very good at what I do. I don’t need others to tell me what is real, good, or valid. I create what I want to create and study what I want to study because that’s what I want to do. That’s the beauty of both art and mathematics—one is free to create their own way. If others accept it, then that may make it more popular or more useful, but it doesn’t make it right. If no one else accepts it, that doesn’t make it wrong. My art makes me happy; it fulfills my need, it keeps me up all hours, it is my passion.

My art--it’s my joy of apathy. I hope you find yours, if you haven’t already.

Untitled (zebra scallop/clamshell)

This image was created using the same formula that I used for the Guilloche Spiral. From the inside out, the boundary curves are rose curves of increasing numbers of petals, from 1 to 20. Between each pair of curves, the spiral completes one turn. Then, I added another series of spirals going the other way, which created a series of discrete areas of roughly concentric rings. I filled every other ring with solid black, making the overall image black on white.

Guilloche Spiral

This image was created with a technique based on guilloche curves—sinusoidal curves bounded by other curves. They are used in engraving as decorations and on banknotes to enhance security. In this technique, I embedded a linear spiral (r = θ) between two curves. The spiral is tangent to the inner curve at its beginning and to the outer curve at its end.

Here, I used a series of boundary curves, from the center outward: a point, a circle, a three-lobed rose curve, a five-lobed rose curve, a circle, and a square. The curves were sized such that each subsequent curve was tangent to the one before it (except for the inner circle). The effect was intended to be reminiscent of DaVinci’s “Vitruvian Man.” Two spiral arms suggest any of the two dichotomies that confront man: male & female, life & afterlife, good & evil, or maybe even the square & circle that surround DaVinci’s man. One arm begins on the right side in the center and the other ends on the left side in the center.

What is This?

This is a place for me to put down into words my thoughts about various aspects of my algorithmic art. It isn't meant for anyone else, but if others wish to read and comment, that's fine. Mainly, it's meant to motivate me to get writing and keep writing. Hopefully, some or all of these musings will make it into a book or an article someday.