Klein Bottle

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) – or, as Leo Moser would put it:

A mathematician named Klein

Thought the Möbius band was divine.

Said he: “If you glue

The edges of two,

You’ll get a weird bottle like mine.”

There are four known parametrizations of the Klein Bottle:

  • The bottle-shape,
  • the four-dimensional, non-intersecting parametrization,
  • the three-dimensional pinched torus,
  • and the figure-8 immersion.

The last is the one presented here, achieved with the following properties:

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


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 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.