How to transform a crosscap into a Boy surface (right-handed or left-handed, your choice) via the Roman surface of Steiner.
September 27 – October 25, 2003
page 3
Plate 8: We begin by moving two cuspidal points (C2 and C4) closer to the triple point T. Dotted lines have then been placed on portions of the surface that will be "hollowed out from the inside" using a "pyramidal punch (please, build these models, otherwise you're headed for the psychiatric hospital). As they develop, the tips of these pyramids are none other than the cuspidal points C2 and C4, which "migrate" and come together.
Plate 9: The cuspidal points meet at S and "annihilate" each other. The self-intersection curve loses two cuspidal points and gains... a loop (in the polyhedral model: a closed polygonal contour).
Plate 10: This "tube with a square cross-section" is formed.
Plate 11: We rotate this object to view it from another angle and move two new cuspidal points, then hollow out the dotted portions "from the inside" (which is nonsensical since the Roman surface of Steiner, a quartic surface, is one-sided). We continue this migration and confluence of this second pair of cuspidal points.
In the last image, the points are close to meeting. Plate 12: The passage is open. Only two cuspidal points remain.
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