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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 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 equation x T ℓ = 0 calculates the inner product of two column vectors. The inner product of two vectors is zero if the vectors are orthogonal. In P 2, the line between the points x 1 and x 2 may be represented as a column vector ℓ that satisfies the equations x 1 T ℓ = 0 and x 2 T ℓ = 0, or in other words a column vector ℓ that is ...
Klein surfaces were introduced by Felix Klein in 1882. [ 1 ] A Klein surface is a surface (i.e., a differentiable manifold of real dimension 2) on which the notion of angle between two tangent vectors at a given point is well-defined, and so is the angle between two intersecting curves on the surface.
Gluing the circles together will produce a new, closed manifold without boundary, the Klein bottle. Closing the surface does nothing to improve the lack of orientability, it merely removes the boundary. Thus, the Klein bottle is a closed surface with no distinction between inside and outside.
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).
A parametric surface need not be a topological surface. A surface of revolution can be viewed as a special kind of parametric surface. If f is a smooth function from R 3 to R whose gradient is nowhere zero, then the locus of zeros of f does define a surface, known as an implicit surface. If the condition of non-vanishing gradient is dropped ...
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 ...