The Klein Bottle Has No Inside or Outside, and Probably Got Its Name From a Pun

A Klein bottle is, in topology, the simplest example of a closed non-orientable surface — a surface that has no inside and no outside, no boundary, and no consistent definition of "left." Walk along it once, all the way around, and you'll find yourself flipped left-for-right at the end. The Klein bottle was first described by the German mathematician Felix Klein in 1882. He coined the German term Kleinsche Fläche — "Klein's surface." The story most topologists tell is that an early translator or typesetter misread Fläche ("surface") as Flasche ("bottle"), and the English-speaking world inherited an apparently unmotivated bottle. Klein's published name was probably surface; the name we use is, almost certainly, a centuries-old typo.

The most striking feature is that the Klein bottle cannot be embedded in three-dimensional space without intersecting itself. The familiar glass-blown sculpture you've seen — with the neck looping through the side wall — is a Klein bottle in 3D immersed, but the self-intersection is real, not a representational shortcut. To get a self-intersection-free Klein bottle you need four spatial dimensions; in 4D you can lift the neck around the wall instead of through it.

The surface plays a particular role in coloring problems. The four-color theorem states that any flat map can be properly colored with four colors. The Heawood conjecture extends this to surfaces of higher genus, and predicts that maps on a Klein bottle would require seven. This turns out to be the only case where Heawood's formula fails: maps on a Klein bottle can always be properly colored with six. Topologically, the Klein bottle is also the connected sum of two real projective planes; cut it along the right plane and it falls apart into two mirror-image Möbius strips.