Preview of a walkable Mandelbrot landscape

You may have seen the Mandelbrot set, but have you seen it from the perspective of an ant walking its slopes? I’m working on a program to provide that perspective; if you’d like a (very early) taste, make sure you’re using an up-to-date Google Chrome browser, and visit this link:

http://www.seethefractals.com/fractaland/index_rel01.html

Elevation-based colors, real-time magnification and borderless exploration are all in the works. The idea is to make it possible to explore the familiar Mandelbrot set fractal from an unfamiliar perspective and see how tasks like finding a minibrot are affected.

Time to escape?

The Mandelbrot set is an escape-time fractal, so called because the color of a given point is determined based on how quickly that point’s orbit escapes some threshold (e.g., a circle of radius=4) when iterated through the Mandelbrot set equation. The Mandelbrot set is a mathematical fractal, as contrasted with natural fractals.

Hypothesis: There are fractal patterns in nature that are of the escape-time variety.

To test this hypothesis, we can look for natural phenomena where, on one level, we see distinct components tracing recognizable orbits through some phase space (similar to the orbits of a single point near the boundary of the Mandelbrot set), and then if we step back, we see a larger fractal pattern being defined by the products of the distinct components (similar to the appearance of the familiar Mandelbrot set shape when many single points near the edge of the set are calculated). Some phenomena which seem good candidates for investigation: Bodies in a solar system, cells in multicellular organisms, and organisms in a society.

Places to look

Benoit Mandelbrot noticed the fractal geometry of certain aspects of nature (hence his book). The front cover of Mandelbrot's book.Science has since discovered the existence of fractal patterns throughout diverse natural phenomena, in fields ranging from cosmology to biology to crystallography to economics. In some cases, the patterns are readily apparent in physical structures such as fern fronds and mountains. In other cases, the fingerprint of fractal geometry may be hiding in temporal structures, such as fluctuations in the stock market. Where else — and how else — might we need to look to discover fractals in natural phenomena that, at first glance, don’t seem to show any signs of exhibiting any aspects of self-similarity?

Awakening

Good morning! I’d like to share with you how to see the fractal patterns in the world around us. Fractal patterns occur on many different levels in nature, and some are more difficult to see than others… but I believe that learning to see the fractals is a first step to understanding what they are and what they might signify.

Let me come clean right now: I hold no advanced degrees in mathematics, physical, or biological science; my background is environmental science and geographic information systems (GIS). I welcome input and feedback from mathematicians and those who hold advanced degrees in the sciences, but such folk are not my primary audience. My aim will be to write in such a way that most folks will be able to “get it”. You’ll have to let me know how successful I am.

So once again, good morning. It’s a going to be a great day.
 

The Point

In James Cameron’s movie “Avatar”, the Na’vi have a saying that translates to “I see you.” Sounds simple, but by the end of the movie the words have acquired some depth of meaning. There’s seeing, and then there’s seeing. Same goes for fractals… just ask Jason Padgett.

Preparing Level 0: Defined.