# Klein Bottle

The Klein bottle is a closed non-orientable surface with neither inside nor outside. Originally presented by Felix Klein, it can be produced by gluing both pairs of opposite edges of a rectangle together giving one pair a half-twist Weisstein (2016). It can be cut into a Möbius strip.

There are two known immersions of the Klein Bottle into real space. The figure-8 immersion is the one presented here, achieved using the following parametrization:

$% \begin{split} x & = \left[\ a + \cos \left(\ \frac{1}{2} u \right)\ \sin v - \sin\left(\ \frac{1}{2}u\right)\ \sin{2v} \right]\ \cos u \\ y & = \left[\ a + \cos \left(\ \frac{1}{2} u \right)\ \sin v - \sin\left(\ \frac{1}{2}u\right)\ \sin{2v} \right]\ \sin u \\ z & = \sin \left(\ \frac{1}{2}u \right)\ \sin v + \cos \left(\ \frac{1}{2}u \right)\ \sin 2v \\ \end{split}$

## References

Weisstein, Eric W. 2016. “Klein Bottle.” Text. http://mathworld.wolfram.com/KleinBottle.html.

## Production details

All prints are produced on a inkjet printer on Innova’s IFA-22 paper:

• 315 g/m2
• Soft textured
• 100% cotton
• Natural white
• Acid and lignin free

Prints are currently only available in the DIN A4 format. If you are interested in other formats, please let me know.

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