The Klein bottle is a closed nonorientable surface of Euler characteristic 0 that has no inside or outside, originally described by Felix Klein. It can be constructed by gluing both pairs of opposite edges of a rectangle together giving one pair a half-twist, but can be physically realized only in four dimensions, since it must pass through itself without the presence of a hole. Its topology is equivalent to a pair of cross-caps with coinciding boundaries. It can be represented by connecting the side of a square in the orientations illustrated in the right figure above (Gardner 1984, pp. 15-17; Gray 1997, pp. 323-324).

cross-cap | Etruscan Venus surface | Franklin graph | Heawood conjecture | Ida surface | Klein bottle crossing number | map coloring | Möbius shorts | Möbius strip