How to transform a Cross Cap surface
into a Boy surface (left or right, at your choice)
passing through the Roman surface of Steiner.
Italian
: Andrea Sambusetti, University of Rome
September 27
- October 25, 2003
page 2
Here is a "Cross Cap surface"
(as you might have discovered it in virtual reality images). It features
two cusp points that are vertices of a self-intersection line.
You can construct it by pinching a balloon with hairpin clips.
But you can also build polyhedral representations. The one below
will particularly interest us.
In table 4 lies the most difficult thing to learn. I find it impossible that anyone could truly understand these objects just by looking at the figures. Build physical models. In simple terms, pull the cusp point
C 2
toward the "inside of the surface" (which, by the way, makes no sense since, as you surely noticed immediately, the Cross Cap surface is one-sided: it has no external and internal faces). By persisting, the surface "self-penetrates," and the self-intersection set is completed, slightly rounded, into an 8-shaped curve. Incidentally, a triple point
Thas been created.
The surface becomes more comprehensible in its polyhedral form, and below we have enlarged certain elements to show what leads us to transform this object into the Steiner Roman surface (see the virtual reality simulation), whose simplest polyhedral form consists of assembling four cubes (here only three are visible).
Table 5: polyhedral version on the left, rounded version on the right. The arrow passes through the point we are going to "pinch." Further down, the beginning of the pinching operation.
Table 6: the pinching is carried out and creates a singular point
B. Indeed, since we pinch it from both sides (to save time), two singular points
S1 and
S1 form, followed by two cusp points. At this point, without Bristol board, scissors, and adhesive tape, you're in trouble.
Table 7: here we have simply moved the different cusp points around. If point
C2 is "obvious," you'll likely have more difficulty identifying points
C 3 and C
4 as cusps. Yet they are there, at the ends of a self-intersection line. Above point
C3 lies simply what I've called a "posicone," that is, a point where positive curvature concentrates (a point where negative curvature concentrates I call a "negacone"). By slightly deforming this object, we arrive at the polyhedral form of the Steiner Roman surface (invented by Steiner in Rome; see its virtual reality illustration).
So the trick is done. There are various types of surfaces, depending on the rules one imposes. Surfaces that do not self-intersect are called "embeddings" (of the sphere or torus in
R3). When they do self-intersect but the tangent plane varies continuously without degenerating, they are called
immersions
. For example: the Klein bottle in its classical representation. In
R 3
, there is no embedding representation of the Klein bottle: it necessarily self-intersects. Immersions have self-intersection sets
without cusp points. These sets are continuous curves, but they may cross at double or triple points. Observation: the sphere can be realized as an immersion (not an embedding) by letting it self-intersect. This is actually the method used to turn it inside out (cf. A. Phillips' method, 1967, whose central step is the double covering of a Boy surface; see also B. Morin and J.P. Petit, 1979, in which the central model is Morin's "four-ear" model, whose polyhedral representation I invented about ten years ago, shown below).
Assembly plan for this object using paper and scissors
If we extend the rules of the game by allowing these objects to include cusp points, we obtain
summersions
(the Cross Cap, the Steiner Roman surface).
I don't know whether "summersion" is the correct term, but since I haven't found any mathematician to clarify this for me, I found it amusing to invent one provisionally, at least until an expert geometer comes forward. Thus, the Cross Cap surface and the Steiner Roman surface would both be summersions of the "projective plane."
To be honest, after twenty-five years of activity and my disappointments in magnetohydrodynamics, I had begun these works because they seemed the furthest possible from any military application. But, as my old friend Mihn pointed out, the term "summersion" might cause confusion and suggest to the Navy that I was trying to conceal progress in submarine propulsion.
The rule of "creation-destruction" of cusp point pairs allows us to pass from one summersion of an object to another, which is exactly what we've just done, showing that the Cross Cap and the Steiner Roman surface are two summersions of the same object, known as the
projective plane.
Don't try to visualize a "projective plane." This object can only be understood through various different representations. As for the term "projective," it's just one among the thousand invented by mathematicians to deter those who wish to enter their closed circle. Zanichelli won't help you at all in mathematics.
We now have to see how to proceed to the Boy surface, which is an immersion of the projective plane.
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