The Mating Dance of the Alexander Horned Spheres
The concept for this Demonstration comes from Tim Poston's cartoon of the same title in the occasional publication Manifold. Move the slider to grow the horns on two Alexander horned spheres. As they grow, they dance or twist around each other until inextricably entangled. Zoom in on the complicated limit point. Control the maximum number of times the horns divide. The performance slider controls how many segments make up each horn. Depending on the number of horns, selecting higher quality can result in slow performance in rendering the image.
Alexander's horned sphere is a famous example, due to J. W. Alexander (1924), of a subset of space homeomorphic to a sphere whose exterior is not simply connected. In this construction each sphere is formed by recursively growing horns, each interlocking not only with its pair on the same sphere but also with the corresponding horn on the sphere's mate. To get completely mated spheres, the construction would have to be carried out to infinity. That is not really possible, but this Demonstration can help get you started.
The original drawing by Tim Poston may be seen here [accessed December 23, 2009].
Note: You may rotate the 3D image as usual, but zooming can be done only by the slider. Also, panning does not work as expected, because of the way the program controls the image presented.
J. W. Alexander, "An Example of a Simply Connected Surface Bounding a Region Which Is Not Simply Connected," Proc. N. A. S., 10(1), 1924 pp. 8–10.