Klein Bottle

The Klein bottle is a closed nonorientable surface of Euler characteristic 0 (Dodson and Parker 1997, p. 125) that has no inside or outside, originally described by Felix Klein (Hilbert and Cohn-Vossen 1999, p. 308). 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 (Francis and Weeks 1999). 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).

It can be cut in half along its length to make two Möbius strips (Dodson and Parker 1997, p. 88), but can also be cut into a single Möbius strip (Gardner 1984, pp. 14 and 17).

The above picture is an immersion of the Klein bottle in (three-space). There is also another possible immersion called the "figure-8" immersion (Apéry 1987, Gray 1997, Geometry Center). While Gray (1997) depicts this immersion using the eight curve (aka. lemniscate of Gerono), it can also be constructed using the usual (Bernoulli) lemniscate (Pinkall 1985, Apéry 1987).

For a fixed abstract Klein bottle , the smooth immersions form four regular homotopy classes (Hass and Hughes 1985, p. 103). If reparametrizations are also allowed, two immersions are identified when is regularly homotopic to for some diffeomorphism . There are three classes under this equivalence (Pinkall 1985, pp. 421-422 and 431). Two of these three classes are interchanged by reflection in , while the third is preserved (Séquin 2013).

The equation for the usual immersion is given by the implicit equation

(1)

(Stewart 1991). Nordstrand gives the parametric form

			(2)
			(3)
			(4)

The "figure-8" form of the Klein bottle is obtained by rotating a figure eight about an axis while placing a twist in it, and is given by parametric equations

			(5)
			(6)
			(7)

for , , and (Gray 1997).

The image of the cross-cap map of a torus centered at the origin is a Klein bottle (Gray 1997, p. 339). The Möbius shorts are topologically equivalent to a Klein bottle with a hole (Gramain 1984, Stewart 2000).

Any set of regions on the Klein bottle can be colored using six colors only (Franklin 1934, Saaty and Kainen 1986), providing the sole exception to the Heawood conjecture (Bondy and Murty 1976, p. 244). The 12-vertex graph (top figures) whose embedding on the Klein bottle (bottom figure) divides it into regions having a minimal coloring using six colors is known as the Franklin graph.