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of Strips and Straps: Open, Closed and Mobius |
This is an investigation of the usefulness of the Euler-Poincaré characteristic for describing such topological figures as the Mobius strip. The conclusion is that the Euler-Poincaré characteristic is useless for characterizing the topological nature of the Mobius strip. There are however some interesting aspects to this investigation.
An open strip is just a rectangle, a two dimensional figure with one face, four edges and four vertices. Its Euler-Poincaré characteristic is therefore 1-4+4=1.
A strip can be closed by bringing one edge into coincidence with its opposite. If the joining is done without twisting the strip then the closed strip has one face, three edges and two vertices. Its Euler-Poincaré characteristic is then 1-3+2=0. This is the same as a torus or other geometric figures with one hole.
If the joining edge and its vertices are erased the result is a cylindrical strip with one face, two edges and no vertices. Its Euler-Poincaré characteristic is then 1-2+0=−1. There may be some problem with an edge having no boundary.
If when an open strip is closed the edge of the strip is twisted 180° then the result is called a Mobius strip. The closed and twisted strip with a joining edge has one face, three edges and two vertices. Therefore its Euler-Poincaré characteristic is 1-3+2=0, the same as for an untwisted closed strip.
If the joining edge is erased there is no change in the Euler-Poincaré characteristic. The edge becomes continuous so the number of edges is one rather than two. The Euler-Poincaré characteristic is then 1-1+0=0.
The essential aspect of the Mobius strip is that it is one sided, as can be established by drawing a pencil line on it. The line can connect any two points on the Mobius strip without crossing an edge. The two dimensional strip however just has one face by virtue of its two dimensionality. To capture the one-sided and two-sidedness of figures a three dimensional version of the strip is needed. This can be called a strap.
An open strap is topologically just a stretched cube and thus has six faces, twelve edges and eight vertices. Its Euler-Poincaré characteristic is thus 6-12+8=2, the same as for any polyhedron.
If the strap is closed (without a twist) then one face with its four edges and four vertices disappears. The closed strap with the joining face retained has five faces, eight edges and four vertices. Its Euler-Poincaré characteristic is therefore equal to 5-8+4=1. If the joining face is erased the result is a cylindrical strap (a ring) with four faces (two vertical and two horizontal), four edges and no vertices. Its Euler-Poincaré characteristic is therefore 4-4+0=0.
When the open strap is closed with a 180° twist and the joining faces and their edges and vertices erased the resulting Mobius strap has two faces, one "vertical" and one "horizontal." It has two edges and no vertices. Therefore its Euler-Poincaré characteristic is 2-2+0=0.
Without the erasure of the joining faces the figure has three faces, six edges and four vertices. Therefore its Euler-Poincaré characteristic is 3-6+4=1. This is the same as the closed strap without a twist. The Euler-Poincaré characteristic does not distinguish between the twisted and untwisted closed straps. Therefore it is useless for capturing the topological aspect of one-sidedness.
(To be continued.)
For more on polyhedra and thire Euler-Poincare characteristic see Euler Poincare.
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