In continuity of Möbius strip we present in this post the Klein Bottle surface.
First of all we give a visualization.
We note that this a two dimensional manifold that is not oriented as the strip we descriped in the previous post. It has taken its name by the German mathematician Felix Klein who was the first one to describe it. Note that , also, Klein bottle has no boundary . This is clearly by its construction.
The Klein bottle, proper, does not self-intersect. Nonetheless, there is a way to visualize the Klein bottle as being contained in four dimensions. By adding a fourth dimension to the three-dimensional space, the self-intersection can be eliminated. Gently push a piece of the tube containing the intersection along the fourth dimension, out of the original three-dimensional space.
More formally, the Klein bottle is the quotient space described as the square $[0,1] \times [0,1]$ with sides identified by the relations $(0, y) \sim (1, y)$ for $0\leq y \leq 1$ and $(0, x) \sim (1-x, 1)$ for $0\leq x \leq 1$.
What is strange about the two surfaces (the strip and the bottle) is that Klein Bottle is a closed manifold , meaning that is a compact one without boundary while the strip has boundary. One more difference is that while the Möbius strip can be embedded in the three dimensional Euclidean space, $\mathbb{R}^3$ , the Klein bottle cannot. That last one, can be embedded however in $\mathbb{R}^4$.
What is the relation , however, of Klein bottle and Möbius strip? Well, if someone dessects it into halves along its plane of symmetry , this would result in two mirror image Möbius strips; one with a left-handed half-twist and the other with a right-handed half-twist.
The Klein Bottle can be parametrized as follows:
$$x = R \left ( \cos \frac{\theta}{2}\cos u - \sin \frac{\theta}{2}\sin 2u \right ) , \;\;\;\;
y= R \left ( \sin \frac{\theta}{2}\cos u + \cos \frac{\theta}{2}\sin 2u \right ) $$
$$z = P \cos \theta \left ( 1 + \epsilon \sin u \right ), \;\;\;\; w = P\sin \theta \left ( 1+ \epsilon \sin u \right )$$
whereas $R, P$ are constants that determine aspect ratio and $\theta$ and $u$ are the angle around the $xy$ plane and $u$ is the little $u$ dependant "bump" to the fourth $w$ axis at the intersection respectively.
First of all we give a visualization.
We note that this a two dimensional manifold that is not oriented as the strip we descriped in the previous post. It has taken its name by the German mathematician Felix Klein who was the first one to describe it. Note that , also, Klein bottle has no boundary . This is clearly by its construction.
The Klein bottle, proper, does not self-intersect. Nonetheless, there is a way to visualize the Klein bottle as being contained in four dimensions. By adding a fourth dimension to the three-dimensional space, the self-intersection can be eliminated. Gently push a piece of the tube containing the intersection along the fourth dimension, out of the original three-dimensional space.
More formally, the Klein bottle is the quotient space described as the square $[0,1] \times [0,1]$ with sides identified by the relations $(0, y) \sim (1, y)$ for $0\leq y \leq 1$ and $(0, x) \sim (1-x, 1)$ for $0\leq x \leq 1$.
What is strange about the two surfaces (the strip and the bottle) is that Klein Bottle is a closed manifold , meaning that is a compact one without boundary while the strip has boundary. One more difference is that while the Möbius strip can be embedded in the three dimensional Euclidean space, $\mathbb{R}^3$ , the Klein bottle cannot. That last one, can be embedded however in $\mathbb{R}^4$.
What is the relation , however, of Klein bottle and Möbius strip? Well, if someone dessects it into halves along its plane of symmetry , this would result in two mirror image Möbius strips; one with a left-handed half-twist and the other with a right-handed half-twist.
The Klein Bottle can be parametrized as follows:
$$x = R \left ( \cos \frac{\theta}{2}\cos u - \sin \frac{\theta}{2}\sin 2u \right ) , \;\;\;\;
y= R \left ( \sin \frac{\theta}{2}\cos u + \cos \frac{\theta}{2}\sin 2u \right ) $$
$$z = P \cos \theta \left ( 1 + \epsilon \sin u \right ), \;\;\;\; w = P\sin \theta \left ( 1+ \epsilon \sin u \right )$$
whereas $R, P$ are constants that determine aspect ratio and $\theta$ and $u$ are the angle around the $xy$ plane and $u$ is the little $u$ dependant "bump" to the fourth $w$ axis at the intersection respectively.
Source: Most Wiki
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