Definition:Klein Bottle
Jump to navigation
Jump to search
Definition
A Klein bottle is a $2$-manifold obtained from the square:
- $\set {\tuple {x_1, x_2} \in \R^2: \size {x_1}, \size {x_2} \le 1}$
by identifying the edges $x_1 = \pm 1$ and $x_2 = \pm 1$ such that:
- $\tuple {-1, x_2}$ is identified with $\tuple {1, -x_2}$ for all $x_2$
- $\tuple {x_1, -1}$ is identified with $\tuple {x_1, 1}$ for all $x_1$.
That is, by identifying both pairs of opposite sides, one with the other, of a square, while twisting the square for one pair:
Also see
- Results about the Klein bottle can be found here.
Source of Name
This entry was named for Felix Christian Klein.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Klein bottle
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Klein bottle