Mobius surface

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. 

The transformation from P to P is therefore not an identity transformation, but rather an involution, with P and F as conjugate points. The identity transformation corresponds to a double rotation of 27t along the Mobius surface. The two sides of the paper corresponds to a double covering of the non-orientable topological Mobius surface. 

The symbols x and y indicate that the surface remains open perpendicular to this edge. Closure of the Mobius surface can be done in one of two ways, according to the plane diagrams, (a) and (b) shown overleaf. 

The idea that transition states can be of Mobius type, in which the relative stability of AN- and 4A- -2-electron systems is reversed, was developed and systematized by Zimmerman [33], who derived the Woodward-Hoffmann Rules for the various thermo-rearrangements in terms of the Hiickel or Mobius nature of their transition states. As shown in (a) and (b) of Fig. 1.4, a cycloaddition in which one of the reaction partners reacts suprafacially and the other antarafacially mimics a Mobius surface, so [t 2s +7r2a]-cycloaddition is allowed whereas reaction along the [ Aa pathway is forbidden, as is the... 

A Mobius surface is easily constructed by pasting the ends of a rectangular strip of paper to one another top-to-bottom, rather than top-to-top as one would to form an ordinary ring. 

In Eqs. (27) and (28), p is the contribution of the substrate water molecules, p that of the adsorbate polar head, and p that of the hydrophobic moiety of the adsorbed molecules. Consistently, 8i, 82, and 83 are the effective local permittivities of the free surface of water and of the regions in the vicinity of the polar head and of the hydrophobic group, respectively. The models have been used in a number of papers on adsorbed monolayers and on short-chain substances soluble in water. " Vogel and Mobius have presented a similar but more simplified approach in which p is split into two components only. " Recently some improvements to the analysis using Eq. (27) have been proposed. " An alternative approach suggesting the possibility of finding the values of the orientation angle of the adsorbate molecules instead of local permittivities has been also proposed.""... 

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. ... 

Just as on the Mobius strip, strange things would happen if we lived on the surface of a small hypersphere. By analogy, consider a Flatlander living in a universe thar is rhe surface of a small sphere. If the Fladander travels along the... 

Dhathatthereyan A, Baumann U, Muller A, Mobius D (1988) Biochim Biophys Acta 944 265 13Z Horvolgyi Z, Nemeth S, Fendler J (1993) Colloids and Surfaces A 71 327... 

For a homoantiaromatic system, surface delocalization in the cyclopropyl ring is parallel to the bridging bond, thus forming a Mobius antiaromatic electron ensemble delocalized along the periphery of the bi(poly)cyclic ring system. 

Edited by T. Kajiyama and M. Aizawa Vol. 5 Foam and Foam Films. By D. Exerowa and P.M. Kruglyakov Vol. 6 Drops and Bubbles in Interfacial Research. Edited by D. Mobius and R. Miller Vol. 7 Proteins at Liquid Interfaces. Edited by D. Mobius and R. Miller Vol. 8 Dynamic Surface Tensiometry in Medicine. By V.M. Kazakov, O.V. Sinyachenko,... 

Vol. 11 Novel Methods to Study Interfacial Layers. By D. Mobius and R. Miller Vol. 12 Colloid and Surface Chemistry. By E.D. Shchukin, A.V. Pertsov, E.A. Amelina and A.S. Zelcnev... 

When a cyclic polyene is large enough, it can exist in both cis- and iraws-forms. Our approach to polyene cyclization has tacitly assumed an all cis -n chain in the form of a band or ribbon that would slip smoothly on to the surface of a cylinder of appropriate diameter. Should the orbitals of the two polyenes in (36) have a mismatch in their orbital symmetries, a single twist in the tt band of one of them could remedy this (Fig. 10c). Cycloaddition would now be allowed and the reaction would proceed, provided other factors were favorable. Such cases of Mobius (Zimmerman, 1966), anti (Fukui and Fujimoto, 1966b) or axisymmetric (Lemal and McGregor, 1966), as opposed to Hiickel, syn, or sigma-symmetric ring closure are unknown (or, at least, rare). A Mobius form has, however, been proposed as the key intermediate in the photochemical transformations of benzene (Farenhorst, 1966) in (48) in place of the disrotatory cyclization proposed by van Tamelen (1965). 

Additionally, if only S0 and Sx are considered, these configurations cannot mix as a consequence of Brillouin s theorem. Nevertheless, the situation is not as serious as it appears. Thus it is seen that in an MO approximation the S0 and Si configurations do become degenerate where HOMO-LUMO crossing occurs, and this signifies that where Mobius-Huckel theory predicts a degeneracy, surfaces will at least approach one another. 

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. 

An imperfect lower-dimensional analogue of the envisaged world geometry is the Mobius strip. It is considered imperfect in the sense of being a two-dimensional surface, closed in only one direction when curved into three-dimensional space. To represent a closed system it has to be described as either a one-dimensional surface (e.g. following the arrows of figure 7) curved in three, or a two-dimensional surface (projective plane) closed in four di-... 

Figure 7.7 Chirality is gradually inverted on transplantation along the surface of a Mobius strip. The same hand is shown on opposite sides of the double cover.

So far, all the surfaces considered are two-sided. In order to pass from a point on one side of a given surface to a point on the other side it is necessary to cross a boundary curve. If the surface is closed, then it would be necessary to penetrate the surface in some way. Not all surfaces, however, are two sided. Because of the half-twist required to form a Mobius strip, the resulting surface is one-sided. It is now possible to travel from any point originally on one side of the strip to any point originally on the other side, without crossing a boundary curve. As shown in figure 14 a Mobius strip has only one boundary curve, since another effect of the half-twist is that the... 

Unlike other closed surfaces the Mobius strip is bounded. The boundary is a simple closed curve, but unlike an opening in the surface of a sphere it cannot be physically shrunk away in three-dimensional space. When the boundary is shrunk away the resulting closed surface is topologically a real projective plane. In other words, the Mobius strip is a real projective plane with a hole cut out of it. 

It is a one-sided surface. That is, any point P on it can be joined to its opposite, Q (or to any other point) by a path that does not cross the edge of the surface. It is named after the nineteenth century mathematician, August Mobius. 

If a Mobius strip is cut length-wise the result will be just one two sided surface. If cut again, the result will be two interconnected surfaces that, again, are two sided. 

A Mobius strip is a twisted surface in space that is made by starting with a rectangular piece of paper, twisting one side through 180° (relative to the opposite side), and then joining it to the opposite side. That is. 

The equation applies only to orientable surfaces, those with distinct sides. This excludes one-sided surfaces, such as the Mobius strip.) Thus, a sphere has genus zero. A torus (or a sphere with one handle) has genus one, and so on. 

L. Collins-Gold, D. Mobius, and D. G. Whitten, Interfacial effects on excited-state potential energy surface. Interrelationship between photoreactiviry and surface properties, Langmuir 2, 191-194 (1986). 

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