Klein bottle

The Klein Bottle went the Möbius Strip one better. It is the closed (i.e. compact without boundary) non-orientable surface with Euler characteristic = 0. It may be obtained by attaching two Möbius bands along their boundary circles. The first step forms a tube, but the second step can not be carried out without causing self-intersection: the tube must pass through itself in order to attach the ends correctly. It can be physically realized only in 4-D, since it must pass through itself without the presence of a hole.

It can be cut in half along its length to make two Möbius Strips, but can also be cut into a single Möbius Strip.

It has no inside or outside. You can see one at www.kleinbottle.com.

Also, any bottle found in the possession of Alberta Premier Ralph Klein. Empty, or imminently so.

A mathematician named Klein
Thought the möbius strip was divine.
He said, "If you glue
The edges of two
You get a weird bottle like mine."

You can get 3D approximations of a Klein bottle (with a hole) from http://www.kleinbottle.com/; you can also see how to knit them, without a hole (from a certain point of view) at http://web.meson.org/topology/.

You can buy real, honest to goodness Klein Bottles from Acme Klein Bottles Inc. at www.kleinbottle.com! This is a one-man operation by Cliff Stoll (yup that guy).

Don't be discouraged that the webpage looks weird, the klein bottles (or a 3D immersion thereof) really are for sale! He's also selling Klein Hats, Klein Steins, and portraits of Gauss ($10 Deutschmark bills).

Every topologist should have one.

A Klein bottle is a 4-dimensional object that has no true inside or outside, much like a mobius loop. In fact, the conception of a Klein bottle, created by Felix Klein, came from the idea of sewing two mobius loops together. A normal bottle has a 'crease' or 'fold': where the opening of the bottle is. This is where the inside and outside of the bottle meet. A sphere does not have this crease or fold, but it has no opening. A Klein bottle has an opening, but not a crease: it is a continuous structure. Because it has no crease or fold, it has no verifiable definition of where it's 'inside' and 'outside' begin. Therefore, the volume of a Klein bottle is considered zero, having no real contents.

Like a mobius strip, the surface of a Klein bottle is continuous. There is no inside and outside, but simply one side. Both sides of the glass are actually on the same side! This is due to the way the bottle twists upon itself, making one continuous surface, rather like a spiral twisted in the shape of a bottle. If this is hard to visualize, it is because it is impossible to exist in 3-D dimensional space. If it did exist, the bottle would have to pass through itself somewhere in order to only have one side, which would create a crease, thereby making it not a continuous surface and not a Klein bottle.

Another one of the properties of Klein bottles is they are non-orientable. This means a flat object, such as a penny, can be pulled across its surface and, without picking the penny up or flipping it, could return to the same point on the bottle laying on it's other side. This property is not present in spheres or regular bottles, or any other real object, so they are orientable.

Although Klein bottles can't truly exist, they can be 'immersed' into three dimensions. This creates an 'almost' Klein bottle, with many of the same properties, including one side and one hole.

A parametric representation of a Klein bottle, with 'little' radius (the radius of the tube) R₁ and 'big' radius (the radius of the loop the tube makes) R₂, is as follows:
x = (R₁ * cos(t))
y = (R₁ * sin(t) * sin(u / 2))
z = ((R₂ * cos(u)) + (R₁ * sin(t) * cos(u / 2)))
w = (R₂ * sin(u))
(0 ≤ u ≤ (2 * π))
(0 ≤ t ≤ (2 * π))

The parameterisation also works with the intervals:
(0 ≤ u ≤ (4 * π))
(0 ≤ t ≤ π)

To make a 3D model, add the w value to the y-coordinate, this will make the shape a bit clearer.