The Klein Bottle is named after Felix Klein, who conceived the idea of joining two Möbius Loops together, thereby generating a single sided tube with no boundary; its outside is its inside, and vice versa.

To form a Klein Bottle, you join the opposite sides of a rectangle to make a cylinder. Then you connect the other pair of sides with a half-twist; unfortunately, this isn’t possible in our three-dimensional universe. An authentic Klein Bottle can only exist in four-dimensions because the surface must pass through itself without creating a hole (a little difficult to visualize).

The image shown is the ‘usual’ configuration, which is an immersion in three-dimensions; apparently there is another one possible, the figure-eight model. The equations (for those beyond common curiosity) can be found here.

animated image found here