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