# Klein bottle (Robert Israel version, z- coordinate)

## Description

n mathematics, the Klein bottle is an example of a non-orientable surface, informally, it is a surface (a two-dimensional manifold) in which notions of left and right cannot be consistently defined. A Klein bottle has no boundary. The “figure 8” immersion (Klein bagel) of the Klein bottle has a particularly simple parametrization. It is that of a “figure-8” torus with a 180 degree “Möbius” twist inserted. In this immersion, the self-intersection circle (when v = 0, π) is a geometric circle in the xy-plane. Robert Israel version is a parameterization of the 3-dimensional immersion of the Klein bottle.

Related formulas## Variables

z | Z-coordinate (dimensionless) |

u | Rotation angle ( 0 ≤ u < π ) (radians) |

v | Rotation angle ( 0 ≤ v < 2π) (radians) |