Definition:Klein Bottle

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