Τριαλάβαστρον (Trialábastron), a triple Klein bottle, is an artistic exploration of Klein bottle geometry. In topology, the Klein bottle is a non-orientable surface that cannot be embedded in three-dimensional Euclidean space without self-intersection, though it can be smoothly realized in four dimensions.
A close relative is the Möbius strip, another non-orientable surface in three-dimensional space. On such surfaces, a continuous path can return to its origin with reversed orientation, revealing how geometry can twist the notion of inside and outside.
Here is another representation of a triple Klein bottle—UK sculptor Alan Bennett’s striking construction of three Klein bottles nested within one another. Of course, “inside” only applies in the sense of their embedding in physical space; from a topological standpoint, they remain entirely separate, non-interacting surfaces.
