Klein Bottle

A Klein bottle, described by Felix Klein in 1882, is a closed non-orientable surface with no boundary and no distinguishable inside and outside. In four-dimensional space it can be embedded without any self-intersection, but the familiar three-dimensional rendering must pass the surface's neck through its side, producing an apparent self-intersection that is purely an artifact of the lower-dimensional projection. The Klein bottle is closely related to the Möbius strip: gluing two Möbius strips along their boundary circles produces a Klein bottle.

The parametric formula uses u and v from 0 to 2\pi with a scale parameter a = 3, producing the standard immersion in three dimensions. The glass material and grounded scene emphasize the tube-like geometry, including the narrow neck that loops back through the wider body. Walking along the surface eventually returns the walker to the start mirrored — the hallmark of non-orientability.