Mobius strip opening

Figure 2.8 The relationship between the four spirals of figure 7 is best understood by mapping on the double cover of a Mobius strip, opened up in this diagram.

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. 

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. 

Walba et al. extended their conception to the synthesis of belt-shaped crown ethers by lengthening the open-chained precursor 32 by one unit [29]. The cyclization of the diol-ditosylate 36 was carried out analogously to the method cited above and gave two products with the same molecular weight as determined by Cl mass spectrometry in 50% overall yield. As CPK models suggest the four functional groups of 36 can react with each other not only parallel but also crossed . This leads to the cylinder 37 and the first molecular Mobius strip 38. A Mobius strip as an isomer of a cylinder has only one surface and one edge. 

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