Properties
The Klein bottle can be seen as a fiber bundle over S1 as follows: one takes the square (modulo the edge identifying equivalence relation) from above to be E, the total space, while the base space B is given by the unit interval in y, modulo 1~0. The projection π is then given by π = .
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 constructed (in a mathematical sense, because it cannot be done without allowing the surface to intersect itself) by joining the edges of two Möbius strips together, as described in the following anonymous limerick:
- A mathematician named Klein
- Thought the Möbius band was divine.
- Said he: "If you glue
- The edges of two,
- You'll get a weird bottle like mine."
Six colors suffice to color any map on the surface of a Klein bottle; this is the only exception to the Heawood conjecture, a generalization of the four color theorem, which would require seven.
A Klein bottle is equivalent to a sphere plus two cross caps.
When embedded in Euclidean space the Klein bottle is one-sided. However there are other topological 3-spaces, and in some of the non-orientable examples a Klein bottle can be embedded such that it is two-sided, though due to the nature of the space it remains non-orientable.
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