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# Klein bottle

Where a Möbius strip is a two dimensional object with only one surface and one edge, a Klein bottle is a two dimensional object with a single surface, and no edges. Consider for comparison, that a sphere is a two dimensional surface with no edges, but that has two surfaces.

A Klein bottle can be constructed by taking a rectangular subset of $\mathbb{R}^{2}$ and identifying opposite edges with each other, in the following fashion:

Consider the rectangular subset $[-1,1]\times[-1,1]$. Identify the points $(x,1)$ with $(x,-1)$, and the points $(1,y)$ with the points $(-1,-y)$. Doing these two operations simultaneously will give you the Klein bottle.

Visually, the above is accomplished by the following. Take a rectangle, and match up the arrows on the edges so that their orientation matches:

This of course is completely impossible to do physically in 3-dimensional space; to be able to properly create a Klein bottle, one would need to be able to build it in 4-dimensional space.

To construct a pseudo-Klein bottle in 3-dimensional space, you would first take a cylinder and cut a hole at one point on the side. Next, bend one end of the cylinder through that hole, and attach it to the other end of the clyinder.

A Klein bottle may be parametrized by the following equations:

$\displaystyle x$ | $\displaystyle=\begin{cases}a\cos(u)\bigl(1+\sin(u)\bigr)+r\cos(u)\cos(v)&0\leq u% <\pi\\ a\cos(u)\bigl(1+\sin(u)\bigr)+r\cos(v+\pi)&\pi<u\leq 2\pi\end{cases}$ | ||

$\displaystyle y$ | $\displaystyle=\begin{cases}b\sin(u)+r\sin(u)\cos(v)&0\leq u<\pi\\ b\sin(u)&\pi<u\leq 2\pi\end{cases}$ | ||

$\displaystyle z$ | $\displaystyle=r\sin(v)$ |

where $v\in[0,2\pi],u\in[0,2\pi],r=c(1-\frac{\cos(u)}{2})$ and $a,b,c$ are chosen arbitrarily.

## Mathematics Subject Classification

54B15*no label found*

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