How to transform a Cross Cap surface
into a Boy surface (right or left, your choice)
passing through the Roman Steiner surface.
Italian
: Andrea Sambusetti, University of Rome
**September 27
- October 25, 2003**
page 3
Plate 8: We begin by making two cuspidal points (C2 and C4) migrate, bringing them a little closer to the triple point T. To do this, we have highlighted with dots a part of the surface that we will "break through from the inside," with a "pyramidal awl" (come on, build some models, otherwise you're ready for the madhouse). As they develop, the tips of these pyramids are none other than the cuspidal points C2 and C4 that migrate and come together.
Plate 9: the cuspidal points join together at S and "cancel each other out." The self-intersection curve therefore loses two cuspidal points and gains... a ring (in polyhedral form: a closed polygonal line).
Plate 10: this "tube with a square cross-section" forms.
Plate 11: we rotate this object to see it from another point of view, and we make two other cuspidal points migrate, after which we break through "from the inside" (which is absurd, since we said that the Roman Steiner surface is one-sided) as before the parts highlighted with dots. We continue with this operation of migration-confluence of this second pair of cuspidal points.
In this last image the points are about to touch. Plate 12: the passage between the two small pyramids has opened up. Now only two cuspidal points remain.
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