A Klein bottle, named after the German mathematician Felix Klein (1849-1925), is a one-sided curved topographical surface. As a single sided surface it has no inside or outside. Hence, it cannot contain volume.
If the fact that a Klein Bottle has no inside or outside reminds you of a Möbius strip, there is good reason. Take a full slice of the Klein Bottle and you have a Möbius strip. There is even a rhyme devoted to the relationship between the two.
A mathematician named Klein
Thought the Möbius band was divine.
Said he: "If you glue
The edges of two,
You'll get a weird bottle like mine."
The name Klein Bottle is derived from an error in the interpretation of the German word Fläche, meaning surface. This was taken to be the word Flache or bottle - and the name stuck. Today we call it the Klein Bottle in English, and Kleinsche Flasche in German.
A fine video on this surface, produced for a topology seminar at the Leibniz Universität in Hannover, Germany, is found below.
Of course, a true Klein Bottle cannot be made in three-dimensions (no intersections allowed). Consider this to be a snap-shot of the 'real thing'.
Handcrafted in the U.S.A.
Dimensions: 4 to 4.5 inches (11-12 cm) high, about 2 inches (5.5 cm) in diameter.
Displacement: about 5 fl. oz. (140 ml).
Actual volume: precisely 0 fl. oz. (0.0 ml).
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