Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Klein bottle
Summary
In mathematics, the Klein bottle (ˈklaɪn) is an example of a non-orientable surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. More formally, the Klein bottle is a two-dimensional manifold on which one cannot define a normal vector at each point that varies continuously over the whole manifold. Other related non-orientable surfaces include the Möbius strip and the real projective plane. While a Möbius strip is a surface with boundary, a Klein bottle has no boundary. For comparison, a sphere is an orientable surface with no boundary.
The Klein bottle was first described in 1882 by the mathematician Felix Klein.
The following square is a fundamental polygon of the Klein bottle. The idea is to 'glue' together the corresponding red and blue edges with the arrows matching, as in the diagrams below. Note that this is an "abstract" gluing in the sense that trying to realize this in three dimensions results in a self-intersecting Klein bottle.
To construct the Klein bottle, glue the red arrows of the square together (left and right sides), resulting in a cylinder. To glue the ends of the cylinder together so that the arrows on the circles match, one would pass one end through the side of the cylinder. This creates a curve of self-intersection; this is thus an immersion of the Klein bottle in the three-dimensional space.
Image:Klein Bottle Folding 1.svg
Image:Klein Bottle Folding 2.svg
Image:Klein Bottle Folding 3.svg
Image:Klein Bottle Folding 4.svg
Image:Klein Bottle Folding 5.svg
Image:Klein Bottle Folding 6.svg
This immersion is useful for visualizing many properties of the Klein bottle. For example, the Klein bottle has no boundary, where the surface stops abruptly, and it is non-orientable, as reflected in the one-sidedness of the immersion.
The common physical model of a Klein bottle is a similar construction. The Science Museum in London has a collection of hand-blown glass Klein bottles on display, exhibiting many variations on this topological theme.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.