How to transform a crosscap into a Boy surface (right or left, as desired) via the Roman surface of Steiner.
September 27, October 25, 2003
page 3
Slide 8: We begin by migrating two cusp points (C2 and C4) toward the triple point T. We then mark dashed lines on portions of the surface that will be "punched from the inside" using a "pyramidal punch" (please, build these models yourself, otherwise you're headed for psychiatric hospital. As the pyramid tips expand, they are none other than the cusp points C2 and C4, which "migrate" and meet each other).
Slide 9: The cusp points meet at point S and "annihilate" each other. The self-intersection curve loses two cusp points and gains... a loop (in polyhedral terms: a closed polygonal contour).
Slide 10: This "square-section tube" is formed.
Slide 11: We rotate the object to view it from another angle and migrate two new cusp points, then punch the dashed parts "from the inside" (which is pointless, since a Roman surface of Steiner, a surface of degree four, is unilaterial). We continue this migration-confluence of the second pair of cusp points.
In the final image, the points are nearly joined. Slide 12: The passage is open. Only two cusp points remain.
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