The planar acetyl groups lie close to planes perpendicular to the helix axis, and project at angles of 120° apart. This allows the chains to hydrogen-bond side by side in a variety of trigonal and hexagonal arrangements, invariably trapping columns of water molecules. The... [Pg.401]

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-... [Pg.237]

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. [Pg.243]

In contrast to the orthorhombic benzene I structure in which the molecules adopt an approximately cubic close-packed arrangement, the molecular arrangement in benzene II is more like hexagonal close packing in other words, this crystal contains definite layers of molecules (in the projection plane... [Pg.16]

The only way in which to transform Minkowski space into a closed manifold is by adding a point at infinity to each coordinate axis, to produce a multiply-connected non-orientable hypersurface, known as a projective plane. General relativity should therefore ideally be formulated as a field theory in projective, and not in affine, space. [Pg.13]

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. [Pg.17]

Figure 3.30 Closing a hemisphere into a projective plane as shown by two-... |

As the total field is inferred closed in both the Z, as well as the Z/N directions, the Mobius model is incomplete and should be expanded into a projective plane, which cannot be embedded in 3-dimensionaJ space. Like the physical imiverse, the cosmic distribution of matter should then also be specified in fom--dimensional space-time. The reconstruction of Figures 5.4 and 5.7 can therefore, at best, be seen as a three-dimensional caricatme of the actual fom--dimensional distribution in the curved Minkowski space of general relativity. [Pg.155]

The reasonable, but not essential assumption, that the general curvature of space-time be constant, predicts a closed topology in the form of either a hypersphere or a four-dimensional projective plane. Additional evidence is needed to decide between these possibilities. [Pg.302]

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. [Pg.304]

Figure 6.13 Schematic (110) projection of the 635X34015 structure. The horizontal lines refer to Ba03 close-packed planes. |

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