In- and Outside the Torus

The torus shown in "In- and Outside the Torus" is the projection of a torus on the three-sphere in four-space projected into three-space. The projection point is on the torus itself, so the projected surface seems to stretch out to infinity

Here we see the torus rotating around a vertical axis. You can see through the "handle" of the torus as it rotates. There is also a tube running vertically.

The region in "front" of the torus is congruent to the region "behind" it. In this movie, the surface rotates around an axis that interchanges the front and back regions. This axis actually lies within the surface itself, and is one of the four circles on the torus that has been projected to a straight line.

Here we rotate the surface around one of the other straight lines on the projected surface. This line is harder to see, since the bands cut across it, unlike the axis in the previous movie, but the rotation still interchanges the front and back regions, showing that they are congruent.

The previous movies all showed rotations in three-space, but this one shows the surface rotating in four-space. The initial image is a torus of revolution, but as the torus rotates in four dimensions, it seems to expand on one side (the side that moves closer to the projection point) and contract on the other. As the torus passes through the projection point, its image in three-space appears to extend to infinity, and it turns inside out: the region that was outside is now inside, and vice versa. The rotation continues and the torus passes through the projection point several more times before returning to its original position.