A Klein bottle

Figure 5.9 Computer graphic representation of a Klein bottle. (Computer rendition by the author see Appendix F for program code.)...

Imagine your frustration (or perhaps delight) if you tried to paint just the outside of a Klein bottle. You start on the bulbous outside and work your way down the slim neck. The real 4-D object does not self-intersect, allowing you to continue to follow the neck that is now inside the bottle. As the neck opens up to rejoin the bulbous surface, you find you are now painting inside the bulb. 

If an asymmetric Flatlander lived in a Klein bottle s surface, he could make a trip around his universe and return in a form reversed from his surroundings. Note that all one-sided surfaces are nonorientable. Figure 5.10 is a glass Klein bottle created by glassblower Alan Bennett (see note 2 for more information). Figure 5.11 is a more intricate Klein-bottle-like object. 

The following is a fragment of code from a C program that draws little spheres along the surface of a Klein bottle. 

Since chirality is a geometrical property, all serious discussions on this topic require a mathematical treatment that is much out of this review. Note, however, that if you cut by the middle of a Klein bottle (an achiral object having a plane of symmetry), you obtain two Mobius strips both chiral and mutually enantiomorph (Fig. 3.5). This pure mathematical result is closely related to the situation of meso compounds described above [11]. 

Mobius strip. In this way the antimatter mystery disappears matter and antimatter are one and the same thing, which merely appear to be different depending on their position in the double cover. In more dimensions the Mobius model is replaced by a projective plane, obtained from an open hemisphere on identifying points on opposite sides of the circular edge. Topologically equivalent constructs are known as a Roman surface or a Klein bottle. 

Figure 18. A Klein bottle drawn to show its relationship to the torus. It has one surface (there is no "inside and outside ) which intersects with itself (Figure 19). Whereas a torus can be formed by Joining the two ends of a cylinder end-face to end-face, a Klein bottle is the result of joining it end-face behind end-face. It can also be formed by joining together the perimeters of two Mobius bands.

Neither of these operations is possible in three dimensional space, but feasible in four. The first of these (a), where all vertices are joined to form a closed surface, is known as a Klein bottle. It can be constructed by a... 

Type 5. We denote by a Klein bottle and by the space of an oriented skew product of by a segment, that is, = K xD The boundary of is a torus T. FVom the topological point of view, the manifold is not essentially new either, because (see the proof below) it is represented as the following gluing ... 

A torus transforms into a Klein bottle (covering it twice) and "vanbhes from the level surface of the integral /. The notation b — K — 0. 

A Klein bottle . Here Nc = fc and is homeomorphic to a Klein bottle... 

When a system does not have a global cross-section, the unstable manifold of the saddle-node may also be a Klein bottle (if the system is defined in... 

Theorem 12.3. (Afraimovichr-Shilnikov [3, 6]) If the global unstable set of the saddle-node L is a smooth compact manifold a torus or a Klein bottle) at fi = Oy then a smooth closed attractive invariant manifold 7 (fl torus or a Klein bottle, respectively) exists for all small fi. 

If m = 1, then is a manifold. It is homeomorphic to a torus if m 1 and to a Klein bottle if m = — 1. As already mentioned, this manifold may be smooth or non-smooth, depending on whether intersects the strong-stable foliation transversely everywhere or not. When x and (p are... 

Chapter 12. Global Bifurcations at the Disappearance of... 12.3. The formation of a Klein bottle... 

Let us consider next the bifurcation of the saddle-node periodic orbit L in the case where the unstable manifold is a Klein bottle, as depicted in Fig. 12.3.1, i.e. when the essential map has degree m = -1. By virtue of Theorem 12.3, if is smooth, then a smooth invariant attracting Klein bottle persists when L disappears. In its intersection with a cross-section So, the flow on the Klein bottle defines a Poincare map of the form (see (12.2.26))... 

Virtual bifurcations of such kind were named the blue sky catastrophes by R. Abraham. The first example of a blue sky catastrophe was constructed by Medvedev [95] for the saddle-node bifurcation on a Klein bottle. The most important feature of Medvedev s example is that the periodic orbit whose length and period are constantly increasing as /i -hO remains stable and does not undergo any bifurcation for all small /x > 0. Theorem 12.8 shows that this is only possible in the case fo (p) = 0, which means that all points (except for the two fixed points) of the essential map are of period two. 

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