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Pendulum Tori

As we have seen in In- and Outside the Torus, the three-sphere can be formed by two solid tori glued along their surface. If we consider these tori to be formed by a collection of tori with progressively smaller radii, ending at a circle at the center of each torus, then we have the three-sphere decomposed into a family of tori starting and ending at circles. Because the two solid tori were linked, their central circles are linked, and so are the tori near these circles.

Stereographic projection of one of these tori from the three-sphere into three-space forms a cyclide of Dupin. Here we see two such linked cyclides (red and green). A third one (banded blue) appears between them, and represents one of the infinite family of intermediate tori. In this case, it contains the projection point, so it extends to infinity and is in transition between the tori nested around the red one to those nested around the green one.

The name comes from the fact that these tori can be used a configuration spaces for the physical system known as the double pendulum, which is formed by one pendulum swinging from the end of another pendulum. For a given ratio between the lengths of the two pendulums, the various positions of the system corresponds to points on a fixed torus within the family depicted here.

This image appeared in Beyond the Third Dimension, and is part of the video The Hypersphere: Projections and Foliations.

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Created: 13 Oct 2003
Last modified: Oct 13, 2003 5:49:43 PM
Comments to: dpvc@union.edu
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