Klein Bottles

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

Glass Klein bottle designed and manufactured by glassblower Alan... 

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. 

Mathematica is a technical software program from Wolfram Research (Champaign, Illinois), With this versatile tool it is possible to draw beautiful Klein bottles as discussed in Chapter 5. The following is a standard recipe for creating Klein bottle shapes using Mathematica. 

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

Stewart, I. (1998) Glass Klein bottles. Scientific American. 278(3) 100—101. 

Spherical and toroidal fullerenes have an extensive chemical literature, and Klein bottle polyhexes have been considered, for example, in [Kir97, KlZh97]. 

Figure 3.1 Smallest spherical, toroidal, Klein bottle and projective fullerenes. The first column lists the graphs drawn m the plane, the second the map on the appropriate surface and the third the dual on the same surface. The examples are (a) Dodecahedron (dual Icosahedron), (b) the Heawood graph (dual Ky), (c) a smallest Klein bottle polyhex (dual 3,3,3), and (d) the Petersen graph (dual Ke).

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]. 

EXPERIMENTS IN TOPOLOGY. Stephen Barr. Classic, lively explanation of one of the byways of mathematics. Klein bottles, Moebius strips, projective planes, map coloring, problem of the Koenigsberg bridges, much more, described with clarity and wit. 43 figures. 210pp. 5K x 8b. 25933-1 Pa. 4.95... 

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. 

There are also a number of other fairly obvious stereoisomeric possibilities (i.e., ones transparent to graph theory) that have as yet received no attention, e.g., multiple toroidal rings (or Klein bottles) with a catenane type of interlocking knots tied in the torus tube concentric multilayered toroidal tubes. 

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.

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