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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:
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.
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