The Etruscan Venus

State of the art computers, such as the Silicon Graphics IRIS, have something to offer journeyman topologists that can otherwise be experienced only in the imagination. It is the wonder of animating, in real time, complicated deformations of topological surfaces, interactively! This paper reports a project at the National Center for Supercomputing Applications (NCSA) to visualize a regular homotopy of a Klein bottle immersed in 4-space. The shadow (projection) of this phenomenon in 3-space is a mapping homotopy between stable images of closed, one-sided (non-orientable) 2-manifolds called ovalesques. Such surfaces are generated by the prescribed motion of an oval (e.g. an ellipse) through space. Thus, the notion of an ovalesque is a projective generalization of a ruled surface. Recall that ruled surfaces are generated by straight lines.