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In- and Outside the Torus

This striking image is a projection into three-space of a torus that lies on the three-sphere in four-space. One way to form a three-sphere is to glue two three-balls together on their spherical surfaces; these balls then form the upper and lower "hemispheres" of the three-sphere, and their common two-sphere is the equatorial sphere.

There is another way to make a three-sphere, however, and that is to glue two solid tori together along their surface. This can be done symmetrically within the three-sphere, as can be seen polyhedrally using the hypercube as a model (four of its eight cubes attached end-to-end form one torus, and the other four the other torus). It also has connections to the analysis of functions of two complex variables.

In this image, the boundary torus where the two solid tori are glued is projected stereographically into three-space. The projection point actually lies on the torus itself, so its projection seems to stretch out to infinity, dividing all of three-space into two congruent regions (the projections of the two solid tori). We are inside one and outside the other. The boundary torus is banded by Hopf circles, and alternate bands are removed to help make the structure more visible.

The sequence of projections as the torus rotates in four-space is shown on the TFBCON2003 home page. This image appeared as part of the art show Surfaces Beyond the Third Dimension, where its mathematics are discussed in more detail.

[Link] In- and Outside the Torus in SB3D

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Created: 13 Oct 2003
Last modified: Oct 15, 2003 10:55:15 AM
Comments to: dpvc@union.edu
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