In and Outside the TorusThis striking image is a projection into threespace of a torus that lies on the threesphere in fourspace. One way to form a threesphere is to glue two threeballs together on their spherical surfaces; these balls then form the upper and lower "hemispheres" of the threesphere, and their common twosphere is the equatorial sphere. There is another way to make a threesphere, however, and that is to glue two solid tori together along their surface. This can be done symmetrically within the threesphere, as can be seen polyhedrally using the hypercube as a model (four of its eight cubes attached endtoend 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 threespace. The projection point actually lies on the torus itself, so its projection seems to stretch out to infinity, dividing all of threespace 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 fourspace 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.


