A.4 Eye Candy
The thing that draws people to fractals are the images. The following are collections of those I particularly like.
Software for producing these images can be found in Appendix 2: Software Resources.
Strange attractors exhibit infinite degrees of layering. What look like lines in a two dimensional attractor turn out to be groups of lines, those lines are themselves groups of lines, and so on. Three dimensional attractors have a similar structure, but are made up of an infinite number of infinitely thin layers, like a well made pastry crust.
Zoom Into the Mandelbrot Set
The boundary of the Mandelbrot set, like all fractal objects, shows equal amounts of detail at all levels of magnification. The following set of images show the details found at ever greater magnifications up to the computational limit of my computer.
|M = 10−1||M = 100||M = 101||M = 102|
|M = 103||M = 104||M = 105||M = 106|
|M = 107||M = 108||M = 109||M = 1010|
|M = 1011||M = 1012||M = 1013||M = 1014|
Wander Aimlessly Through the Mandelbrot Set
Portions of the Mandelbrot set are similar to each other, but not in the manner of other fractal objects. Whereas objects like Sierpinski's triangle, Koch's coastline, Peano's monster curve, and even Julia sets exhibit strict self-similarity; the Mandelbrot set is filled with motifs that resemble each other, but never quite seem to repeat. Bulbs, branches, filaments, and even miniature copies of the Mandelbrot set itself crop in weird and interesting variations everywhere you look.
Lyapunov Space Diagrams
The following Lyapunov space diagrams show the stability of the driven logistic equation. The logistic equation is a simple model for predicting the size of animal populations from year to year based on their fecundity (something like a combination of reproduction and survival rates). These particular diagrams show a population driven by rates "a" and "b" alternating in a periodic sequence. The Lyapunov exponent is a measure of the stability of a population. On a Lyapunov space diagram, stability is shown from white (least stable) to dark gray (most stable), while instability is shown as black regardless of degree.