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A two-dimensional representation of the Klein bottle immersed in three-dimensional space. In mathematics, the Klein bottle (/ ˈ k l aɪ n /) 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.
In 1961, Professor Cardwell asked Mitsugi Ohno to construct a true glass Klein bottle, a one-sided figure formally described as “an enclosure continuous with its outer surface constructed by twisting a tube through an opening in the side of the tube and joining it to the other end". Klein bottles had previously been made by other glassblowers ...
In mathematics, a solid Klein bottle is a three-dimensional topological space (a 3-manifold) whose boundary is the Klein bottle. [ 1 ] It is homeomorphic to the quotient space obtained by gluing the top disk of a cylinder D 2 × I {\displaystyle \scriptstyle D^{2}\times I} to the bottom disk by a reflection across a diameter of the disk.
The Klein bottle, immersed in 3-space. In mathematics , an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective . [ 1 ] Explicitly, f : M → N is an immersion if
The fundamental group of the Klein bottle can be presented in the form a , b ∣ a b a − 1 = b − 1 . {\displaystyle \langle a,\;b\mid aba^{-1}=b^{-1}\rangle .} and is therefore a semidirect product of the group of integers with addition, Z {\displaystyle \mathrm {Z} } , with Z {\displaystyle \mathrm {Z} } .
Felix Christian Klein (German:; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations between geometry and group theory.
This relation holds, as Ringel and Youngs showed, for all surfaces except for the Klein bottle. Philip Franklin (1930) proved that the Klein bottle requires at most 6 colors, rather than 7 as predicted by the formula. The Franklin graph can be drawn on the Klein bottle in a way that forms six mutually-adjacent regions, showing that this bound ...
The Klein bottle can be turned into a Klein surface (compact, without boundary); there is a one-parameter family of inequivalent Klein surfaces structures defined on the Klein bottle. Similarly, there is a one-parameter family of inequivalent Klein surface structures (compact, with boundary) defined on the Möbius strip. [2]