Transformation of the Crosscap into a Boy surface, via the Roman surface of Steiner
How to transform a crosscap into a Boy surface (left or right, as desired) by passing through the Roman surface of Steiner.
**September 27, October 25, 2003 **
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Plate 8: We start by moving two cusp points (C2 and C4) closer to the triple point T. We then draw dotted lines on parts of the surface that will be "pierced from the inside" with a "pyramidal punch (please, build these models, otherwise you are headed for the psychiatric hospital). As they develop, the tips of these pyramids are none other than the cusp points C2 and C4 which "migrate" and come together.
Plate 9: The cusp points come together at S and "annihilate". The self-intersection curve loses two cusp points and gains ... a loop (in polyhedral: a closed polygonal contour).
Plate 10: this "square-section tube" is formed.
Plate 11: we rotate this object to see it from another angle and we move two new cusp points, then we pierce the dotted parts "from the inside" (which is silly since a Roman surface of Steiner, a surface of the 4th degree) is a one-sided surface). We continue this migration-confluence of this second pair of cusp points.
In the last image, the points are close to coming together. Plate 12: the passage is open. Only two cusp points remain.
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