Klein Bottles

The Klein Bottle is an important figure in topology and is an example of a non-orientable surface. It was first described in 1882 by the German mathematician Felix Klein.

Möbius strip and Klein Bottles

The Möbius Strip is the simplest one-sided surface and can easily be made by cutting out a strip of paper, twisting it, and glueing the two ends together.

Source: Wikipedia

It was discovered in 1858 by the German astronomer and mathematician August Ferdinand Möbius.

The Möbius strip is related to the Klein bottle; if you cut the Klein bottle in half along its length you can make two Möbius strips, or it can be cut into a single Möbius strip.

The boundary of this strip is a single, closed curve. To remove its boundary, simply pull all boundary points of the strip continuously together. Whether or not we round off the tip of the cone, a closed surface without a boundary is obtained. This forms a type of Klein bottle – 3D pinched torus / 4D Möbius tube.

Source: Wikipedia

Non-Orientable? One sided?

A one-sided surface is one that, when standing upright, you can walk along the surface and reach both sides of each point of the surface. In nature, most surfaces are two-sided.

A surface is said to be orientable if a shape drawn on it cannot be transformed into its mirror image by moving the shape along the surface. This means that the both the Möbius strip and the Klein bottle are non-orientable.

Other Types of Klein Bottles

Bottle type: This type of Klein bottle can be made by using a cylinder. Instead of adding a twist as we did with the Möbius strip, loop one end back through the cylinder and glue it to the other end by adjusting the thickness to obtain a one sided surface.

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Source: plus.maths.org

Figure 8-Klein bottle: a Klein bottle obtained by rotating a figure 8 about an axis, while placing a twist in it.

Source: Wolfram Alpha

It is described by the parametric equations:

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

where r is the radius of the circle, θ gives the rotation of the figure 8 and v specifies the position around the cross section.

The Lawson-Klein bottle: H.B Lawson created a Klein bottle that is part of the family of helicoidal (stair-like) surfaces.

Source: plus.maths.org

Additionally, the Klein bottle can be projected into 4-D to created a non-intersecting, non-orientable surface.

Numberphile Video

Sources: 1 | 2

Hope you enjoyed this post! M x


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