# Klein Bottle

Klein Bottle
Fields: Geometry and Calculus
Image Created By: 3DXM Consortium
Website: 3D Xplor Math

Klein Bottle

The Klein Bottle is a non-orientable surface with no boundary first described in 1882 by the German mathematician Felix Klein.

# Basic Description

The Klein Bottle is a one-sided, non-orientable surface. Unlike the, more well known, Mobius strip, the Klein Bottle is a surface without boundary. As a result, 3 dimensional models of it intersect, or pass through, themselves.

# A More Mathematical Explanation

The Figure 8 immersion of the Klein bottle can be parametrised with the following equat [...]

The Figure 8 immersion of the Klein bottle can be parametrised with the following equations:

• $x = \left (r + \cos \left (\frac{v}{2} \right ) \sin \left (u \right )- \sin \left (\frac{v}{2} \right ) \sin \left (2u \right ) \right ) \sin \left (v \right )$
• $y = \left (r + \cos \left (\frac{v}{2} \right ) \sin \left (u \right )- \sin \left (\frac{v}{2} \right ) \sin \left (2u \right ) \right ) \sin \left (v \right )$
• $z = \sin \left (\frac{v}{2} \right ) \sin \left (u \right )+ \cos \left (\frac{v}{2} \right ) \sin \left (2u \right )$

For $u$ and $v=[0,2\pi)$