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Picture a bottle with a hole in the bottom. Now extend the neck. Curve the neck back on itself, insert it through the side of the bottle (a true Klein bottle in four dimensions would not require this step, but it is necessary when representing it in three-dimensional Euclidean space), and connect it to the hole in the bottom.
Unlike a drinking glass, this object has no "rim" where the surface stops abruptly. Unlike a balloon, a fly can go from the outside to the inside without passing through the surface (so there isn't really an "outside" and "inside").
Topologically, the Klein bottle can be defined as the square [0,1] × [0,1] with sides identified by the relations (0,y) ~ (1,y) for 0 ≤ y ≤ 1 and (x,0) ~ (1-x,1) for 0 ≤ x ≤ 1, as in the following diagram:
----> ^ ^ | | <----Like the Möbius strip, the Klein bottle is a two-dimensional differentiable manifold which is not orientable. Unlike the Möbius strip, the Klein bottle is a closed manifold, meaning it is a compact manifold without boundary. While the Möbius strip can be embedded in three-dimensional Euclidean space R3, the Klein bottle cannot. It can be embedded in R4, however.
The Klein bottle can be con structed (in a mathematical sense) by joining the edges of two Möbius strips together, as described in the following anonymous limerickThis article is about the poetic form. For other uses of the name, see Limerick (disambiguation). A limerick is a short, often humorous and ribald poem developed to a very specific structure. Structure The rhyme scheme is usually aabba, with a rather rigi:
If a Klein bottle is dissected into halves along its plane of symmetryIn 3-dimensional geometry, a plane of symmetry is a 2-dimensional flat dividing surface placed such that things on one side are symmetrical (mirror image) to things on the other side. The term has been used in organic chemistry to describe the molecules', the result is the surface shown in the following figures.
In Figure B, twenty-six points on the dissection's perimeterThe perimeter is the distance around a given two-dimensional object. For a polygon it is calculated by adding the lengths of all of the sides. For circles the equation is P 2 π r, where r is the radius and π is the mathematical constant. See also: i (the blue curve) have been labeled with the twenty six letters of the alphabetAn alphabet is a complete standardized set of letters—basic written symbols—each of which roughly represents or represented historically a phoneme of a spoken language. This as distinguished from other writing systems such as ideograms, in which symbols r. But the dissection is a surface, not a curve. The red lines show how the surface is subtended by the perimeter.
A Möbius strip, a surface with only one side. Glueing together two Möbius strips, one obtains the Klein bottle.
The Möbius strip is a surface: its perimeter is shown as a blue curve, and the red lines show how the surface is subtended by the perimeter.
In both the dissected Klein bottle and Möbius strip, the red lines connect letters which are related mutually in the ROT13ROT13 ( ROTate by 13 places , sometimes hyphenated ROT-13 sometimes lowercase rot13 is a simple Caesar cipher used for obscuring text by replacing each letter with the letter thirteen places down the alphabet. A becomes N, B becomes O and so on. The algor cipher. This helps to illustrate that half a Klein bottle is homeomorphic to a Möbius strip.
It is also possible to perceive directly that Figure A is a Möbius strip, by imagining that the narrower, re-entrant part of the bottle no longer intersects line segment DB after the dissection is performed, but that it becomes loose from dissecting plane and Figure A is actually three-dimensional, with line segments VW and IJ hovering above line segment DB. Then, suddenly, Figure A looks like a roller coaster, and by imagining the motion of a rail car along the blue rails of this roller coaster, one perceives that this roller coaster is non-orientable.
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