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The initial construction of the Klein bottle by identifying opposite edges of a square shows that the Klein bottle can be given a CW complex structure with one 0-cell P, two 1-cells C 1, C 2 and one 2-cell D. Its Euler characteristic is therefore 1 − 2 + 1 = 0.
In mathematics, a solid Klein bottle is a three-dimensional topological space (a 3-manifold) whose boundary is the Klein bottle. [ 1 ] It is homeomorphic to the quotient space obtained by gluing the top disk of a cylinder D 2 × I {\displaystyle \scriptstyle D^{2}\times I} to the bottom disk by a reflection across a diameter of the disk.
The fundamental polygon of a Klein bottle. To construct the Klein bottle, join the sides with the same letter together so that the directions of the arrows match. (This cannot be done in three dimensions without the surface intersecting itself.)
Klein bottle: . Note that the sphere and the projective plane can both be realized as quotients of the 2-gon, while the torus and Klein bottle require a 4-gon (square). The expression thus derived from a fundamental polygon of a surface turns out to be the sole relation in a presentation of the fundamental group of the surface with the polygon ...
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English: Save this script (slightly modified from - see for the page containing this model) to file "bottle" then run the command line "gnuplot bottle", you will get "bottle.svg", if you have well installed gnuplot 4.0 or later.
When an archival reference did come into play, it was executed in unexpected ways, like a new take on a squared-off mule from fall 1999 or the iconic CK One perfume bottle she transformed into an ...
This corresponds to the fact that in the fundamental polygon of the Klein bottle, only one pair of sides is glued with a twist, whereas in the projective plane both sides are twisted. A square is a contractible topological space, which implies that it has trivial homology. Consequently, additional cuts disconnect it.