Klein bottle

   A
mathematician named Klein,
   Thought the Möbius
band was divine,
   He said, “If you glue,
   The edges of two,
   You’ll get a weird
bottle like mine.”
    
– Leo Moser

Take a rectangle and join one pair of opposite sides to make a cylinder.
Now join the other pair with a half-twist. The result is a Klein bottle.
Sound easy? It is if you have access to a fourth
dimension because that’s what is needed to carry out the second step
and to allow the surface to pass through itself without a hole. A true Klein
bottle is a four-dimensional object. It was discovered in 1882 by Felix Klein when he imagined, as in the limerick,
joining two Möbius bands together to
create a single-sided bottle with no boundary.

An ordinary (three-dimensional) bottle has a crease or fold around the opening
where the inside and outside of the bottle meet. A sphere doesn’t have this crease or fold, but it has no opening. A Klein bottle
has an opening but no crease: like a Möbius band, it is a continuous one-sided
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 to be zero, and the bottle has no real contents –
except itself! As the joke goes: “In topological hell the beer is packed
in Klein bottles.” Take a coin, slide it across the surface of a Klein bottle
until it returns to its starting point, and the coin, as if by magic, will
be flipped over. This is because, unlike a sphere or a regular bottle, a
Klein bottle is non-orientable.

Although a Klein bottle can’t be ’embedded’ (that is, fully realized) in
three dimensions, it can be “immersed” in three dimensions. Immersion is
what happens when a higher dimensional object cuts through a lower dimensional
one, producing a cross section. When a sphere is immersed in a plane, for
example, it produces a circle. A three-dimensional glass model of a Klein
bottle can be made by stretching the neck of a bottle through its side and
joining its end to a hole in the base. Except at the side-connection (the
nexus), this properly gives the shape of a four-dimensional Klein bottle.
Just as a photo of such a bottle is of a three-dimensional Klein bottle
immersion, so the immersion in real life is like a photo of the true four-dimensional
bottle.