Jesús de la Peña Hernández
14 TOPOLOGIC EVOCATIONS
If we think of topology as the science of surfaces continuity, we are taken to Möbius
bands and flexagons, specially studied by Miguel Ángel Martín Monje.14.1 MÖBIUS ́ BANDS
They are paper strips very long as compared with their width, having both extremities
glued to each other.
Let ́s imagine a band with red (R) obverse and white (B) reverse. We shall call disconti-
nuity limit to any of the two long sides of the rectangle for they are the borders between the red
and the white. On the contrary, we ́ll assume surface continuity across the small sides for they
are stuck.
We shall study several cases. For the sake of clarity, the drawings shown are unidimen-
sional.- If we glue the red extremity over the white one without twisting the band (zero twists), we
get a cylindrical surface white in the convex side and red in the concave one. Now, if we
imagine an ant walking forward on the white surface without crossing the discontinuity
limits, it will give an infinity of turns over the same white surface. If we would have placed
the ant on the red surface and made a similar experiment, the ant would walk all the time on
the red surface.
SUMMARY: glue R / B, zero twists, n (even or odd) cuts (parallel to the discontinuity lim-
its and equidistant). RESULT: According to Fig.1, n independent bands, with same length
and configuration as the original, having a width of 1 / n. - R / B; even number of twists (Fig. 2 has two); zero cuts. There is no transition from one col-
our to the other, as in case 1. Length and width are the original. - R / B; 2 twists; one cut. Result in Fig. 3: two interlaced bands having each of them the same
configuration of Fig. 2.- Keeping the left-hand side band of Fig.3 as it is, we produce a cut in the other. Result: the
left band stands fixed whereas the other is divided in two with a length equal to each of Fig.
3; They appear interlaced between them and with the one in the left.
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