Demonstrating Fractal Reflections Between Reflective Spheres

Introduction

When I displayed an interest in reflections and kaleidoscopes, Dr. Noe introduced me to a seemingly simple project consisting of 4 Christmas balls. When I looked into the center of their reflections, the beautiful fractal reflections that appeared quickly raised my curiosity. Upon researching the topic, I slowly began to be pulled in to a still very young topic in science. In 1999, David Sweet, Edward Ott, and James A. Yorke analyzed the multiple reflections that occur between the surfaces of four reflective spheres when these are stacked up in a tetrahedral arrangement ["Topology in chaotic scattering," Nature 399, 315 (May 27, 1999)]. For my project, we created a slightly smaller copy of the model created by Sweet, Ott, and Yorke.

Background

The term "fractal" was coined by Benoit Mandelbrot in 1975. Fractals are geometric curves or shapes that are "infinitely complex." More specifically fractals have the following properties ...

Properties of Fractals:

- Self-Similarity: Each iteration is an identical copy of the previous iteration. I In other words, shapes and colors look the same regardless of magnification. Fractals DO NOT simplify.

- Iteration: To form a fractal you begin with an initiator and iterate using a generator. Fractals tend to be chaotic. In other words, small changes in the initiator can create chaotic changes in results.

- Infinite, yet Finite: Fractals are infinite (either in length, perimeter, or plane), and at the same time finite (either in area or volume). In other words, the area will be finite on any scale, but the function itself goes to infinity. Fractals are infinite, but can be contained within a finite area.

- Wada Basin Property: All 3 basins (in our case, colors) appear in our view regardless of magnification.

- Fractals form in dimensions of a fraction, called the fractal dimension.

Examples of Fractals:

Some other examples include:

The Sierpinski Triangle,

Setup

Our model was made using four 10" reflecting spheres ("gazing globes"), purchased for $30 each at a large garden supplies store. The balls were stacked in a tetrahedral shape with their stems pointing away from the center.

To make the tetrahedron surrounding the spheres we used 3/16" thick Foam board. Each triangle which made up the tetrahedron had 34.64" sides. Three holes (7.875" in diameter) were cut out of each of the triangles, except the base, for light to enter. Then we used cellophane on two of the holes, and black construction paper on the base to distinguish colors.

Results

References