# 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 has no edges, or more technically, boundary. As a surface without boundary, the Klein Bottle is a true 2 manifold. The presence of an edge in the Mobius strip allows sections of the shape to pass alongside each other, thus avoiding self intersection. Because the Klein Bottle has no edges there is no way for the surface to go around itself when it needs to close up. 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 $v=[0,2\pi)$

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