Mobius band

We also considered in Section 1.2 the possibility of preparing Mobius-band molecules from cycloacenes. In fact regular cycloacenes, that is molecules built from joining both ends of linear acenes without twisting, have never been obtained, despite several experimental attempts. They are interesting since they represent the basic cylindrical carbon units of zig-zag ( ,0) nanotubes. 

A Mobius band is an example of a nonorientable space. This means, in theory, it is not possible to distinguish an object on the surface from irs reflected image. The surface is considered nonorientable if it has a path that reverses the orientation of creatures living on the surface, as described in the previous paragraph. On the other hand, if a space preserves the handedness of an asymmetric structure, regardless of how the structure is moved about, the space is called orientable. ... 

The design of this SAR device was oriented on the so-called Mobius band, which is a twisted structure (see Figure 1.123) [141]. Two horizontal fluid layers are separated in the center. Thereafter, both double layers are turned by 90° in the same direction so that two vertically laminated systems are achieved. Then, the layers are joined to give a four-lamellae system. 

Closely allied to the property of one-sidedness is the property of non-orientability. A surface is said to be orientable if the orientation of an object in the surface is preserved. Consider the handed (chiral) object at a point in the Mobius surface of figure 7. From a local point of view there is a corresponding point on the other side of the surface. Since the Mobius band is one-sided it is possible to draw a continuous path connecting the two points without crossing a boundary curve, as in figure 7. The chirality of the object is reversed when moved along the continuous path between the two points. A situation like this is not possible with two-sided surfaces. 

For many years chemists try to obtain twisted molecules resembling the Mobius band. The feasibility of such systems was discussed for the first time by H. Frisch and E. Wasserman as early as in 1961. 

A new aliphatic helical topology was realized with the synthesis of the first molecular Mobius-band molecule 89, which was obtained in 57 % yield from 90. The chirality of 89 was proved by the NMR method using (+)-(2,2,2)-trifluoro-9-anthrylethanol as an optically active solvent... 

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.

The symmetry that combines the different periodic arrangements of the elements is summarized best by mapping to a projective plane, a two-dimensional section of which is a Mobius band. This construct is an attractive model for a closed universe in which the conjugate chiral forms of matter are separated in a natural way. 

To understand the topology of the projective plane P, we start with the two-dimensional form known as a Mobius band. 

In Figure 3.28 let the line AB represent the position at which the two ends of the paper were joined to form the Mobius band. Symbolically this is equivalent to the plane diagram ... 

The real projective plane may also be constituted from a Mobius band and a disc. The boimdary of a Mobius band is a closed curve, topologically equivalent to a circle. It can therefore be imagined attached by its boundary to the boundary of a disc so as to form a closed surface, the real projective plane. A Mobius band may therefore be thought of as the real projective plane with a disc cut out of it. 

The real projective plane, like a Mobius band, is one-sided and non-orientable. Like the Mobius band, which cannot be embedded in two-dimensional space, the deformations needed to produce a real projective plane cannot be performed in ordinary three-dimensional Euclidean space. Quoting Flegg (1974) the merit of non-Euclidean geometries is that they ... 

Suppose that the angle of rotation is (j>. Each point of the disc x is seen to be mapped to some unique point, x, which is the image of no other point. There is only one point, at the centre, that maps to itself for a rotation oi fixed-point theorem states that in this case no hxed point occurs. Such surfaces lack a special point. They have the alternative property that hair on such a surface can all be brushed to lie in the same direction, unlike the hair on a disc, a sphere, or a human head which develops a crown. This is a striking property of a Mobius band, showing that all points in the surface are quivalent and any of these can be considered to be the central point. 

Figure 5.4 signifies more than elemental or nuclide periodicity. It summarizes the appearance of ponderable matter in all modifications throughout the universe. Following the extended hemlines from top left at Z/N = 1.04 — bottom left at 0 —> top right at Z/N = 1.04 bottom right at 0, and back to top left, the involuted closed path, which is traced out, is mapped to the non-orientable surface of a Mobius band in Figure 5.7. The two sides of the double cover are interpreted to represent both matter and antimatter.

In terms of this simple alternative geometry, geodesic transplantation fixes points on the line element to occur along the double cover of a narrow Mobius band. It is a known property of a Mobius band that a point, which moves along the double cover, close to one edge, rotates around the central line without intersecting it. This is what Godel describes as rotation with respect to a compass of inertia. 

The graphical representation of the way in which chemical periodicity varies continuously as a function of the limiting ratio (Figure 5.3), 1 < Z/N < 0, appears strangely unsymmetrical, despite perfect symmetry at the extreme values. By adding an element of mirror symmetry a fully symmetrical closed function, that now represents matter and antimatter, is obtained. To avoid self overlap the graphical representation of the periodic function is transferred to the double cover of a Mobius band, which in closed form defines a projective plane. 

The projection of a Mobius band onto the middle circle is a fiber bundle with the Mobius band being the total space, the base space B = S, and the fiber F = [0,1] this bundle shown on the right of Figure 8.1. 

A mixing unit consists of a twisted band, either left- or right-handed, which is similar to the so-called Mobius band, a ring-shaped structure with a one-sided surface Therefore this type of mixer is called Mobius mixer The principle is illustrated in figure 3 The layers of two non-miscible fluids are separated perpendicular to the boundary layers, subsequently twisted and reunited, thus doubling the exchange... 

The points at Z/A = 1.04 are arranged symmetrically around a central point. On reflection of the entire pattern around the line at Z = 51, a set of closed lines, which cannot be embedded in two dimensions, is generated. Drawn on the surface of a Mobius band, the double pattern is readily identified as representing both matter and antimatter and requires the points at 0 and 102 to coincide. In order to achieve this, the single edge of the Mobius band must be glued to itself to create a closed space which can no longer be embedded in three dimensions and is known as projective space. 

In the same section we give the bifurcation diagrams for the codimension two case with a first zero saddle value and a non-zero first separatrix value (the second term of the Dulac sequence) at the bifurcation point. Leontovich s method is based on the construction of a Poincare map, which allows one to consider homoclinic loops on non-orientable two-dimensional surfaces as well, where a small-neighborhood of the separatrix loop may be a Mobius band. Here, we discuss the bifurcation diagrams for both cases. 

The set of all points of the phase space whose trajectories converge to L as t —> +00 (—cx)) is called the stable (unstable) manifold of the periodic orbit. They are denoted by W and W , respectively. In the case where m = n, the attraction basin of L is Wf. In the saddle case, W is (m + l)-dimensional if m is the number of multipliers inside the unit circle, and is (p + 1)-dimensional where p is the number of multipliers outside of the unit circle, p = n — m — 1. In the three-dimensional Cctse, Wl and are homeomorphic either to two-dimensional cylinders if the multipliers are positive, or to the Mobius bands if the multipliers are negative, as illustrated in Fig. 7.5.1. In the general case, they are either multi-dimensional cylinders diffeomorphic to X S, or multi-dimensional Mobius manifolds. 

Fig 7.5.1. Saddle periodic orbit in R are distinguished by the topology of the stable and unstable invariant manifolds which may be homeomorphic to a cylinder (left) or a Mobius band (right). 

As for the original map (10.3.1) the fixed point O is asymptotically stable when Ik < 0 and is a saddle when Ik > 0. In the latter case the stable and unstable manifolds of O are the manifolds and, respectively. In terms of the Poincare map of the system of differential equations, the corresponding periodic trajectory L is stable when Ik < 0, or a saddle when Ik > 0. Note that in the saddle case the two-dimensional unstable manifold W L) is, in a neighborhood of the periodic trajectory, a Mobius band. 

If all Lyapunov values are equal to zero and the system is analytic, then the center manifold is also analytic, and all points on it, except O, are periodic of period two. This means that for the system of differential equations there exists a non-orientable center manifold which is a Mobius band with the cycle L as its median and which is filled in by the periodic orbits of periods close to the double period of L (see Fig. 10.3.2). 

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