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Three Shiny Balls & Wada Basins


Table Of Contents

Introduction

Mirrors are fun, and mirrors that aren't flat are the most fun of all. You do remember roasting ants in the back yard with Mom's makeup mirror when you were a kid? Right?!? This page is about the mathematical simulation of four spherical mirrors, and the fascinating, fantastically complex reflection patterns they produce. Ray-tracing programs are designed to simulate all sorts of optical effects including reflection, and so we will be using the open source ray-tracing package Povray to simulate our mirrors.... First, we need to get a handle on the geometry of the situation.

Take four spheres of the same radius, and place them so that each sphere touches the other three spheres like so:

Note that the centers of the spheres are at the vertexes of a regular tetrahedron, that is a regular, triangular pyramid in the language of high school geometry. Next, place faces on three of the gaps like so:

For our experiments below, the balls are all perfect mirrors. With our mirror finish balls correctly placed, we are ready to begin our tour of the many wonders of mirrors. Click on the images to view a larger, 800x600, version.

Examples

In this section, we start out simple and continue to the complex. First we simply build the necessary structure, and then we play with it a bit.

For our first image, lets color the faces of the tetrahedron red, green, and blue. We will look straight into the gap formed between the three front most spheres.
That was nice, now lets fill in that last face of the tetrahedron with a yellow triangle, and put the camera inside of the gap.
Hmmmm. One would expect more reflections than what we are seeing. The answer is to bump up Povray's max_trace_level parameter to a higher value. For the rest of our experiments, we will use a value of 16. This parameter controls the number of bounces a light ray can take before it is considered done. The image above becomes rather striking now.

If this is taken to the theoretical extreme, then one can prove mathematically that every epsilon ball contains a point of each color. I'll bet you didn't expect me to throw in that little bit of mathematics :) This kind of thing is known as a "Wada basin". Very kool stuff.

Color is good, but what happens if we make the triangles have a gold finish?
Gold is KOOL! What happens if we make them white?
That is enough of looking at the back most sphere. Lets move our camera near the back most sphere, and point it at the front most triangle.
Very nice. Now lets try it with a gold texture.
Wow! That's quite a bit of complexity inside. Lets take a step back and see what we get from further away:
I don't care for the lack of color on the spheres, and I really don't care for that black triangle in the middle. How can we add some color? Well the spheres are all mirrors! Give them something to reflect. Let's place a gold plane about 50 units away from the whole thing, behind the camera.
How about a white plane.
Let's increase the contrast by moving the plane in closer to the objects.

References

For a good, encyclopedic, introduction to the field of fractal science in general, I strongly suggest "Chaos and Fractals: New Frontiers of Science" by Peitgen, Jurgens, and Saupe. This book is notable because of it's clear presentation and breadth of coverage. It's a great book to have around for the casual reader because it is broken up into semi self-contained sections that one can just pick up and read.

Basins of Wada
Kennedy, J. & Yorke, J.A.
Physica D 51, 213-225, 1991

Fractal basin structure
Takesue, S. & Kaneko, K.
Progr. Theor. Phys. 71, 35-49, 1984

Fractal basin boundaries
McDonald, S.W., Grebogi, C., Ott, E. & Yorke, J.A.
Physica D 17, 125-153, 1985

Basins of attraction
H.E. Nusse, J.A. Yorke
Science 271, 1376-1380 (1996)

© 2009 Mitch Richling