Far Out Fractals – Somewhere Between Two and Three Dimensions

Fractals are both insanely interesting and gloriously beautiful. The universe uses them in everything, from snowflakes, to clouds, to coastlines. Sometimes, the forest looks exactly like the trees.

Fractals unleashed by IBM

In the dim and distant past of 1961, mathematical research scientist Benoit Mandelbrot was working for IBM. His one task seemed simple enough. His firm “was involved in transmitting computer data over phone lines, but a kind of white noise kept disturbing the flow of information — breaking the signal.” His job was to eliminate that noise.

The main thing he found out is that it can’t be done. The reason why opened a whole new branch of mathematics. He’s the one who coined the term “fractal” to describe “self-similar” things which always look pretty much the same, no matter how much you zoom the magnification in or out.

A graph of the noise producing “turbulence” quickly uncovered something fascinating, later a whole class of objects termed “fractals.” Regardless of the scale of the graph, “whether it represented data over the course of one day or one hour or one second, the pattern of disturbance was surprisingly similar. There was a larger structure at work.”

Inspired by French mathematicians Pierre Fatou and Gaston Julia, Mandelbrot became obsessed with the simple equation “z = z² + c.” It maps out “values on the complex plane—where the x-axis measures the real part of complex number and the y-axis measures the imaginary part (i) of a complex number.”

One of the most interesting things about fractals is that they don’t classify neatly into one, two, or three dimensional objects. The contour of the edges are so intricate and complex that measuring your way around the edges gives a different number depending on what size ruler you use. While a point is non-dimensional, a line has one dimension and a square has two. Popping a square up into a cube yields three dimensions.

All those lines and edges are crisp and even. Compare that with something like the coastline of Britain. Calculating the “dimensionality” of an object like a Koch Snowflake shows it’s 1.262 dimensional. Lost in limbo between one and two.

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Dissecting the Mandelbrot set

Mandelbrot’s work soon led to an entire set of fractals named just for him after he sprung the idea on the world in 1982 with “The Fractal Geometry of Nature.” To project an iconic “Mandelbrot” fractal, you simply plot the series zn+1 = zn2 + z0 where z is a complex number. That gives the first set of points on a complex plane, “z0.” Next, “If the series diverges (escapes to infinity) for a particular point z0 then it is colored white (or a color or intensity related to how fast the point escaped).”

“If the point doesn’t escape the point is shaded black and is said to be inside the Mandelbrot set.” Fancy software can supply all sorts of psuedo color to highlight the deeper structure. Some of the landmarks are so popular to study they have names.

The Seahorse Valley, for instance, is part of the Mandelbrot set of fractals along the top edge where the spherical curves merge together on the rightmost juncture. Elephant Valley is where the “butt cheeks” are. It’s easy to spend hours cruising into the depths of fractal geometry with modern tools like Xaos. Previously limited to Linux users, they finally released a Windows version and WB highly recommends it.

One of the really freaky places to focus on is the tip of the nose at the far left edge. If you can find it. Every time you zoom in further it extends a little more. Some call it a visualization of what’s happening in the concept called “Kairos” or “fractal time.” Each instant we experience is another bubble inflating around the last one.

The whole core to the concept of fractals is “roughness.” Once you’re exposed to Mandelbrot’s ideas, you instantly start seeing them everywhere. The shapes of mountains, coastlines and river basins for example. Video game programmers were quick to learn that a computer can plot a landscape from a fractal algorithm a whole lot faster and with a fraction of the resources required to paint it in dot by dot.

Benoit himself noted, “The form of geometry I increasingly favored is the oldest, most concrete, and most inclusive, specifically empowered by the eye and helped by the hand and, today, also by the computer … bringing an element of unity to the worlds of knowing and feeling … and, unwittingly, as a bonus, for the purpose of creating beauty.“