Teaching Material

The following material may be helpful in using the spheres to teach students about Chaos.

Setting up the spheres

You should first create a four-spheres setup. You can use four shiny, silver, spherical (ball-shaped) Christmas Tree ornaments to make a small demonstration model, or lawn ornaments to larger demonstration model.

The Small Demo

We want to stack the four Christmas Tree ornaments like cannonballs (thanks to Aaron Nielsen for this comparison), that is, place three on the table or in your hand so that they are each touching the other two. Then place the fourth on top of and in the middle of the three.

You can use transparent tape to hold the balls together (thanks to Dan Lathrop for this idea). If we image that the centers of the balls are on the vertices (corners) of a pyramid, we want to place the tape along the edges of that pyramid. In other words, use the tape to hold each pair of balls together. (I'll post a picture of this soon.)

This demo is fairly stable. I pass it around at talks and poster sessions. Of course, dropping it on the floor will probably shatter the spheres.

The Large Demo

This demo will need to stay put. The lawn ornaments are usually 10"-12" in diameter. The are spherical except for a small cylindrical part that juts out and can be used as a handle.

Before setting up the demo, clean each sphere with windex to ensure the best image. You should also take care to rotate each sphere so that the clearest part of it is facing inward, toward the other spheres. This is where the image will be produced, so we don't want any scratches or deformities in the mirror in there.

When I've set these up I've placed three heavy, metal, hollow, cylidrical canisters on a table placed at the vertices of a triangle. The "handles" on the lawn ornaments fit into the cylinders and the ball is held steady because the cylinder is heavy. The four ball rests on top of the three with no additional support.

Set up this way, the demo is not very stable. You will need to find some way to secure the top ball. You might try using clear packing tape along the edges of the "pyramid" as described in the previous section. If you use this or another method, I'd like to hear about it!

Viewing the Spheres

The fractal

Using either demo, you can shine white light into the spheres -- a flashlight works fine -- and you'll see a fractal image. It will be the bright images that looks like triangles in side of triangles inside of triangles....

The basins

Now, try placing colored paper in front of one of the "holes" through which you have been peering inside the demo. Equivalently, you can imagine that you are placing the paper along on of the faces of the pyramid shape made by the spheres. The paper should be large enough to completely cover the opening.

Now look inside through another opening while shining light onto the colored paper so that the side facing the spheres is illuminated. Notice that many -- but not all -- of the triangles in the image inside the spheres are filled in with the same color as the paper.


D. Sweet, E. Ott, J. A. Yorke
We would like to thank D. Lathrop, B. Zeff, N. Peffley, and E. Boettcher for their assistance.

Information on this page is Copyright (C) 1998-2000 by David Sweet. You may reproduce this page or parts of it for non-profit use as long as the author (David Sweet) is credited in the reproduction. If you use page as a source, it should be properly referenced. For other uses, please contact me at dsweet@chaos.umd.edu. I would prefer that you also include the URL of this page in the reproduction: http://www.chaos.umd.edu/~dsweet/Spheres

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