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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. The bottles date from 1995 and were made for the museum by Alan Bennett. [3] The Klein bottle, proper, does not self-intersect.
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.
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.
For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space. Topological surfaces are sometimes equipped with additional information, such as a Riemannian metric or a complex structure, that connects them to other disciplines within mathematics, such as differential geometry and complex analysis .
The Klein bottle immersed in three ... while four-dimensional Lorentzian manifolds model spacetime in ... A Euclidean vector space with the group operation of vector ...
The Klein bottle, and all other non-orientable closed surfaces, can be immersed in 3-space but not embedded. By the Whitney–Graustein theorem, the regular homotopy classes of immersions of the circle in the plane are classified by the winding number, which is also the number of double points counted algebraically (i.e. with signs).
Lawson's Klein bottle is a self-crossing minimal surface in the unit hypersphere of 4-dimensional space, the set of points of the form ( , , , ) for <, <. [53] Half of this Klein bottle, the subset with 0 ≤ ϕ < π {\displaystyle 0\leq \phi <\pi } , gives a Möbius strip embedded in the hypersphere as a minimal ...
Many hyperbolic lines through point P not intersecting line a in the Beltrami Klein model A hyperbolic triheptagonal tiling in a Beltrami–Klein model projection. In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit ...