Klein Bottle

Klein Bottle

Graphs and Surfaces

Published in Kenneth H. Rosen, Graphs, Algorithms, and Optimization, 2005

The Klein bottle can be constructed by gluing two edges of a rectangle to create a cylinder, and then by gluing the ends of the cylinder together, according to the orientation of the arrows. This also cannot be done in euclidean space, as the cylinder must cross through itself without intersecting itself.

Planarity and Kuratowski’s Theorem

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Published in Jonathan L. Gross, Jay Yellen, Mark Anderson, Graph Theory and Its Applications, 2018

Jonathan L. Gross, Jay Yellen, Mark Anderson

Remark: Surfaces generally fall into two infinite sequences. The three surfaces described above are all relatively uncomplicated and lie in the sequence of orientable surfaces. Surfaces such as the Möbius band and the Klein bottle are examples of unorientable surfaces.

Using design to develop an in-depth understanding of mathematical surfaces: The hairy Klein bottle

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Journal Information

Published in International Journal of Mathematical Education in Science and Technology, 2021

Daniel Cohen, Shai Gul

A Klein bottle is an extension of a Möbius strip, which was first described in 1882 by the German mathematician Felix Klein. It is constructed by joining the edges of two Möbius strips together. The Klein bottle is also non-orientable, but unlike the Möbius strip it does not have an end, i.e. there are no boundary points (Figure 9).

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Graphs and Surfaces

Planarity and Kuratowski’s Theorem

Using design to develop an in-depth understanding of mathematical surfaces: The hairy Klein bottle