Preamble to the article published in 1994 in the journal Nuovo Cimento
The starting point of this work dates back to 1977. Two notes published in the Comptes rendus de l'Académie des Sciences de Paris:
J.P. Petit: "Enantiomorphic Universes with Opposite Time Arrows," CRAS, May 8, 1977, vol. 285, pp. 1217–1221
J.P. Petit: "Universes Interacting with Their Mirror Image in the Time Mirror." CRAS, June 6, 1977, vol. 284, Series A, pp. 1413–1416
In the paper that follows, we attempted to establish a point-by-point (involutory) correspondence between points in the neighborhood of Earth (on a cosmological scale) and their conjugate points in the second universe (which we call the twin universe, shadow universe, or ghost universe—terms we consider equivalent in our mind), using an antipodal relation, which implied an initial assumption about the topology of the geometric object. Later, we realized this was unnecessary, since one could define the local structure (F, F*) as a two-sheeted covering of a "skeletal manifold." The structure then becomes that of a two-sheeted covering of a projective space P3, equivalent to the three-dimensional version of the two-dimensional projective plane P2, which is more familiar. The most well-known representation is the surface discovered in 1902 by the Austrian mathematician Werner Boy (see Figure 184—ideally an animation, once the website is complete).
Boy was a student of the great mathematician Hilbert, who reportedly expressed great satisfaction with his student’s invention. For the record, after inventing this surface, Boy left university and was never heard from again. All historical attempts to trace his whereabouts proved futile. It remains unknown whether he died from a bad flu or ended his days as a plumber.
Geometers know that one can bring all points of a sphere S2 into coincidence with those of a projective plane P2, as illustrated in Figure 10 of the article below. The north pole is thus brought into coincidence with the south pole, and the equator wraps around itself along the pseudo-equator of Boy’s surface, also indicated. This two-sheeted covering is shown in Figure 11 of the article. Note, at least in two dimensions, that this operation brings enantiomorphic objects—mirror images—into coincidence. Figures 12 and 13 are didactic illustrations showing how clumps would then fit into the gaps in the antipodal region.
This two-sheeted covering system can be extended to three and even four dimensions, using spheres S3 and S4, covering respectively the projective spaces P3 and P4.
Before proceeding further, we may help the reader become familiar with the geometry of this strange Boy surface. The reader may also find various variations of the object in the Topologicon (Belin Editions, 1984).
What may surprise the reader is the fact that this surface intersects itself along a set of self-intersections forming a trefoil curve, reminiscent of a ship’s helix:
In this drawing on the left, an opening has been made to reveal the triple point, where three sheets cross. This surface appears quite peculiar. In fact, this object is an excellent example illustrating the concept of a 3D representation space mentioned earlier.
The triple point T and the self-intersection curve arise solely from the way the projective plane P2 is represented in R3. A sphere or a torus can be embedded in R3, meaning they admit topologically equivalent representations where the surface does not intersect itself. However, it is impossible to embed the projective plane P2 in R3. One can only immerse it. The drawing above (Boy’s surface) is therefore an immersion of the projective plane into R3. An immersion of a 2D object is a way of representing it in R3 where one finds a line of double points (the self-intersection curve), along which two tangent planes exist, plus a certain number of triple points where three sheets intersect. Boy’s surface is just one of infinitely many ways to immerse the projective plane P2 into R3. Others can be found in an article to be included on the website, titled "The Different Faces of the Projective Plane."
It is quite easy to generate images of Boy’s surface through a parametric representation we invented and published.
---> The reader will find, in the MATHEMATIQUES subsection of the website, among other things, a reproduction of the note published in 1981 at the Académie des Sciences de Paris, co-authored with J. Souriau (no, not the famous mathematician, but his son Jérôme, who later became a computer scientist), whose reference is:
"Analytical Representation of Boy's Surface," Note aux Comptes Rendus de l'Académie des Sciences de Paris, vol. 293 (October 5, 1981), Series I, pp. 269–272
It shows that the surface has elliptical meridians. This property makes it easy to plot. Below is the program featured on the cover page of my comic book, Topologicon.
BASIC Program
10 CLS
50 PI = 3.14159 : P3 = PI/3 : P6 = PI/8 : P8 = PI/8
90 FOR MU = 0 TO PI STEP .1
95 P = P + 1
100 D = 34 + 4.794 * SIN(6*MU - P3)
110 E = 6.732 * SIN(3*MU - P6)
120 A = D + E : B = D - E
130 SA = SIN(P8 * SIN(3*MU))
140 C2 = SQR(A * A + B * B) : C3 = (4 * D * E) / C2
160 CM = COS(MU) : SM = SIN(MU)
180 FOR TE = 0 TO 6.288 STEP .06
190 TC = A * COS(TE) : TS = B * SIN(TE)
200 X1 = C3 + TC - TS
210 Z1 = C2 + TC + TS
250 REM HERE ARE THE 3 COORDINATES
300 X = X1 * CM - Z1 * SA * SM
310 Y = Y1 * SM + Z1 * SA * CM
350 REM COMMAND TO DISPLAY THE POINTS
360 PSET (X,Y),1
400 NEXT TE : NEXT MU
For the record, it was this discovery of the possibility to represent the surface using elliptical meridians that later enabled the mathematician Apéry to obtain the first implicit, sixth-degree representation:
f(x, y, z) = 0
which we will not reproduce here (it is rather complicated, and we are convinced simpler representations exist—this will be the subject of another document to be included in the MATHÉMATIQUES section of the website).
The Klein bottle is better known to readers. It is also impossible to embed in R3. It appears, in its most classical form, as an immersion with a self-intersection set consisting of a single closed curve.
The two-sheeted covering of Boy’s surface (projective plane P2) is a sphere S2, just as the two-sheeted covering of the Klein surface is a torus T2. Readers interested in Boy’s surface may find a 3D model in one of the rooms at the Palais de la Découverte in Paris, a model we had commissioned from the sculptor Max Sauze based on a more rustic model we had created.
In these two-sheeted coverings, the meridians and parallels of the objects "wrap around themselves." For example, one can illustrate what happens to the "parallels" of the torus (linked to the embedding also shown):
In this embedding of the torus, the parallel curves are obviously not geodesics of the surface (except for the "neck circle"). A similar situation holds for the torus’s meridians, which are geodesics of its embedding...