This movie shows the triple-point surface rotating in space. You can see the Triple-Point from various angles. Several views similar to these appeared as chapter headings in the book Statistics: The Conceptual Approach, by Iverson and Gergen.
Here, the surface is pulled apart so that the triple point is removed. The parameterization of the surface has a constant c that varies from -1 to 1 in this movie. At c = 0, the singularity disappears and the surface becomes embedded.
The triple-point surface is a ruled surface, meaning that is can be swept out by a line moving over time. In this movie, we see this line as it generates the surface. The line first sweeps out a portion of the surface, then seems to backtrack, causing the surface to intersect itself. Finally, it curves back on itself a third time, and passes through the curve of self-intersection again, forming the triple point.
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