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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.