Klein Bottle — Non-Orientable Surface Immersion

The Klein bottle is a non-orientable closed surface with no boundary — it has no inside or outside. It cannot be embedded in ℝ³ without self-intersection. This shows the Figure-8 / Lawson parametrization. Drag to rotate.

Parametrization

Model: U resolution: 40 V resolution: 40 Animation speed: 1.0 Cross-section v₀: 0.00

χ(Klein bottle) = 0
2 Möbius bands glued
Non-orientable: yes
Self-intersects in ℝ³
Embeds in ℝ⁴ cleanly
Drag to rotate