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Klein bottle

Klein bottle
Structure of a three-dimensional Klein bottle In mathematics, the Klein bottle /ˈklaɪn/ is an example of a non-orientable surface; informally, it is a surface (a two-dimensional manifold) in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas 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 German mathematician Felix Klein. It may have been originally named the Kleinsche Fläche ("Klein surface") and that this was incorrectly interpreted as Kleinsche Flasche ("Klein bottle"), which ultimately led to the adoption of this term in the German language as well.[1] Construction[edit] This square is a fundamental polygon of the Klein bottle. By adding a fourth dimension to the three-dimensional space, the self-intersection can be eliminated.

Related:  MATH / GEOMETRYGeometric Topology

One man's Funnies: Mathematical equations of love, heart, penis and the boomerang Love Love is complicated. But the mathematics of it is very simple: - It starts with "I love you" where "I love" is a constant, and "you" is a variable. - Later on, it is: 1 + 1 = 1 - And later still: 1 + 1 >= 3 Any questions? Boy's surface An animation of Boy's surface Boy's surface is discussed (and illustrated) in Jean-Pierre Petit's Le Topologicon.[1] Boy's surface was first parametrized explicitly by Bernard Morin in 1978. See below for another parametrization, discovered by Rob Kusner and Robert Bryant. Boy's surface is one of the two possible immersions of the real projective plane which have only a single triple point. [2]

Real projective plane In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in R3 passing through the origin. (0, y) ~ (1, 1 − y) for 0 ≤ y ≤ 1

Spirographs and the third dimension The basic geometric ideas are straight lines and circles. The famous compass and straight edge. There is a great deal that can be done with just these, but what if you want something more complex? Spirographs are a very simple idea, let one circle run around a second. You can make the circles as cogs and then you get a classic toy. In mathematics there is a mess of names to describe the curves produced, I shall just list them, understanding the differences is a good way of learning the subject: epicycloid, hypocycloid, epitrochoid, hypotrochoid. Borromean rings Mathematical properties[edit] Although the typical picture of the Borromean rings (above right picture) may lead one to think the link can be formed from geometrically ideal circles, they cannot be. Freedman and Skora (1987) prove that a certain class of links, including the Borromean links, cannot be exactly circular. Alternatively, this can be seen from considering the link diagram: if one assumes that circles 1 and 2 touch at their two crossing points, then they either lie in a plane or a sphere. In either case, the third circle must pass through this plane or sphere four times, without lying in it, which is impossible; see (Lindström & Zetterström 1991). A realization of the Borromean rings as ellipses

Curve orientation In mathematics, a positively oriented curve is a planar simple closed curve (that is, a curve in the plane whose starting point is also the end point and which has no other self-intersections) such that when traveling on it one always has the curve interior to the left (and consequently, the curve exterior to the right). If in the above definition one interchanges left and right, one obtains a negatively oriented curve. Crucial to this definition is the fact that every simple closed curve admits a well-defined interior; that follows from the Jordan curve theorem. All simple closed curves can be classified as negatively oriented (clockwise), positively oriented (counterclockwise), or non-orientable.

4-manifold In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure and even if there exists a smooth structure it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic). 4-manifolds are of importance in physics because, in General Relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold. Topological 4-manifolds[edit]

Orientability A torus is an orientable surface In mathematics, orientability is a property of surfaces in Euclidean space measuring whether it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the surface, as needed by Stokes' theorem for instance. More generally, orientability of an abstract surface, or manifold, measures whether one can consistently choose a "clockwise" orientation for all loops in the manifold.

Heart Curve What is the Heart Curve? If you speak about a heart, you rather mean the heart figure than the heart shaped curve. Drawn Heart Curves topMethod 1 Method 2 Method 3 Method 4 Poincaré conjecture By contrast, neither of the two colored loops on this torus can be continuously tightened to a point. A torus is not homeomorphic to a sphere. Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.