Bottle of Klein
In , bottle of Klein is one closed, without edge and nondirectional, i.e. one for which one cannot define a "interior" and a "outside". The bottle of Klein was described for the first time in by the German mathematician . It is closely related to and with plungings of the real projective plan such as surface of Servant boy.
Synopsis |
Construction
The bottle of Klein is not realizable in <maths>\mathbb R^3</maths>, because it is necessary whereas it cross itself ; also, no realization which one can see of the bottle of Klein is exact. In <maths>\mathbb R^4</maths>, it is on the other hand possible to carry it out without car-intersection.
Visualization
It is possible to include/understand the structure of the bottle of Klein starting from the representation provided in this article, and in the price of a mental effort less great than than one could believe.
Let us imagine an alive individual in a flat world, with 2 dimensions. One then tries to explain to the individual what is a n?ud. For that, one draws a n?ud to him on the plan: it sees only one curve which car-is intersected. It is explained to him whereas they are not points of intersection which it sees, but that the curve passes "above" and "below". Our individual is disconcerted: living in a flat world, it does not include/understand what nor is the top what is the lower part. It misses to him a dimension (top and bottom) to be able to visualize the n?ud.
We encounter the same problem when we try to visualize the bottle of Klein, since we see a surface which car-is intersected. Nevertheless, if we reason with the fourth dimension, it is enough to imagine that to this place, the bottle passes "above" and "below" within the meaning of this fourth dimension, and thus car-does not intersect itself.
One can to some extent consider that the bottle of Klein is a surface which makes a "n?ud". As a surface (object with 2 dimensions), it is necessary 4 dimensions for him to make a n?ud, just as for a curve (object with a dimension) one needs 3 dimensions to make a n?ud.
Properties
- The bottle of Klein is the related sum of two real.
- While cutting it into two compared to one symmetry plane, one twice is obtained .
- Its characteristic of Euler-Poincaré is null.
- Its numbers of Betti nonnull are <maths>b_0=1</maths> and <maths>b_1=1</maths>.
- Its chromatic number is 6.
External bond
