image by Archdezart The Klein Bottle The Klein bottle is fascinating. It is a non-orientable surface, which means that the notions of left and right cannot be readily applied. A möbius strip is much like this, but is a surface with a boundary, as seen here. The Klein bottle on the other hand has no boundary, all ‘sides’ are infinitely connected. It was in 1882 that the Klein bottle was first envisioned by the mathematician, Felix Klein. The Klein bottle is interesting in the aspect that unlike a möbius strip, it cannot be embedded (visualised in its entirety) in three dimensional space. However, it can be embedded in 4 dimensional space. If one is to dissect a Klein Bottle along its plane of symmetry, you would find that the results are two mirror image möbius strips. For those who are familiar with parameterization, the Klein bottle has a particularly simple parameterization: a figured 8 torus with a möbius twist. (See below) This so called, Klein Bagel can be graphed through the following equations.     If one were to parametize the Klein bottle in Euclidean space, it becomes much more complex: wherefor 0 ≤ u < 2π and 0 ≤ v < 2π Equations and highly technical information: Wikipedia and various sources

image by Archdezart

The Klein Bottle

The Klein bottle is fascinating. It is a non-orientable surface, which means that the notions of left and right cannot be readily applied. A möbius strip is much like this, but is a surface with a boundary, as seen here. The Klein bottle on the other hand has no boundary, all ‘sides’ are infinitely connected.

It was in 1882 that the Klein bottle was first envisioned by the mathematician, Felix Klein. The Klein bottle is interesting in the aspect that unlike a möbius strip, it cannot be embedded (visualised in its entirety) in three dimensional space. However, it can be embedded in 4 dimensional space.

If one is to dissect a Klein Bottle along its plane of symmetry, you would find that the results are two mirror image möbius strips.

For those who are familiar with parameterization, the Klein bottle has a particularly simple parameterization: a figured 8 torus with a möbius twist. (See below)File:KleinBottle-Figure8-01.png

This so called, Klein Bagel can be graphed through the following equations.

 

\begin{align} x & = \left(r + \cos\frac{u}{2}\sin v - \sin\frac{u}{2}\sin 2v\right) \cos u\\ y & = \left(r + \cos\frac{u}{2}\sin v - \sin\frac{u}{2}\sin 2v\right) \sin u\\ z & = \sin\frac{u}{2}\sin v + \cos\frac{u}{2}\sin 2v \end{align} 

If one were to parametize the Klein bottle in Euclidean space, it becomes much more complex:

\begin{align} x & = \frac{ \sqrt{2} f(u) \cos u \cos v (3\cos^{2}u - 1) - 2\cos 2u}{80\pi^{3}g(u)}-\frac{3\cos u -3}{4}\\ y & = -\frac{f(u)\sin v}{60\pi^{3}}\\ z & = -\frac{\sqrt{2}f(u)\sin u \,\cos v}{15\pi^{3}g(u)}+\frac{\sin u \cos^{2} u + \sin u}{4}-\frac{\sin u\,\cos u}{2} \end{align}where
g(u) = \sqrt{8\cos^2 2u-\cos 2u (24\cos^3 u-8\cos u + 15) + 6\cos^4 u (1 - 3\sin^2 u)+17}for 0 ≤ u < 2π and 0 ≤ v < 2π

Equations and highly technical information: Wikipedia and various sources