Beyond the Third Dimension
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Artist's Comments on In- and Outside the Torus

This image depicts the three-dimensional projection of a torus in four-space. Bands on the torus have been removed to make it possible to "see through" it. Banchoff's description of the torus and the three-sphere provides more information about this image, and includes several MPEG movies of the torus rotating in four dimensions.

This image is an interior view of a cyclide of Dupin, a torus on a three-dimensional sphere in four-space projected stereographically from a point on the torus itself, leading to a third-order algebraic surface expressed as a union of circles (and four straight lines). These curves are orbits of a Hamiltonian dynamical system and the fibers over a great circle of the Hopf mapping from the three-sphere to the two-sphere. The exhibit booklet and the electronic links refer the viewer to several articles written by the author and colleagues in Applied Mathematics and Computer Science, examining different aspects of this extremely important surface. This image is also featured on the cover of the Scientific American Library volume Beyond the Third Dimension.