This image shows a torus that is being sliced by a plane perpendicular to its axis of rotational symmetry, much as a bagle would be sliced. You can think of it as a rising water level enveloping the torus. The slice starts as a single circle, and then splits into two circles which eventually recombine at the top of the torus.

Here the torus is being sliced as though it were being dunked into a cup of coffee. Here the slice curve starts as a circle which becomes "squeezed" at the center, and for a moment is a figure-8. After this, it becomes two circles, which eventually recombine as a figure-8 and then finishes up as a circle.

In this movie, the torus is twisted at a very special angle. Here, the initial circle folds over so that it touches itself at a point. At the same time, its middle squeezes together to touch itself at a second point. Both these touchings occur at the same instant, and at that moment, the slice is formed by two perfect circles. If you have the ability to step through the movie, you might want to do so to see this slice more clearly.

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