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?