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A 6-colored Klein bottle, the only exception to the Heawood conjecture. Six colors suffice to color any map on the surface of a Klein bottle; this is the only exception to the Heawood conjecture, a generalization of the four color theorem, which would require seven. A Klein bottle is homeomorphic to the connected sum of two projective planes. [7]
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.
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.
A chiral molecule is a type of molecule that has a non-superposable mirror image. The feature that is most often the cause of chirality in molecules is the presence of an asymmetric carbon atom. [16] [17] The term "chiral" in general is used to describe the object that is non-superposable on its mirror image. [18]
The mapping class group of the Klein bottle K is: =. The four elements are the identity, a Dehn twist on a two-sided curve which does not bound a Möbius strip, the y-homeomorphism of Lickorish, and the product of the twist and the y-homeomorphism. It is a nice exercise to show that the square of the Dehn twist is isotopic to the identity.
The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. The real projective space RP n is a closed n-dimensional manifold. The complex projective space CP n is a closed 2n-dimensional manifold. [1] A line is not closed because it is not compact.