Klein Bottle

A closed Nonorientable Surface of Genus one having no inside or outside. It can be physically realized only in 4-D (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 cut in half along its length to make two Möbius Strips.

The above picture is an Immersion of the Klein bottle in $\Bbb{R}^3$ (3-space). There is also another possible Immersion called the ``figure-8'' Immersion (Geometry Center).

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

$$(x^2+y^2+z^2+2y-1)[(x^2+y^2+z^2-2y-1)^2-8z^2]+16xz(x^2+y^2+z^2-2y-1)=0$$ (1)

(Stewart 1991). Nordstrand gives the parametric form

$$x = \cos u[\cos({\textstyle{1\over 2}}u)(\sqrt{2}+\cos v)+\sin({\textstyle{1\over 2}}u)\sin v\cos v]$$ (2)

$$y = \sin u[\cos({\textstyle{1\over 2}}u)(\sqrt{2}+\cos v)+\sin({\textstyle{1\over 2}}u)\sin v\cos v]$$ (3)

$$z = -\sin({\textstyle{1\over 2}}u)(\sqrt{2}+\cos v)+\cos({\textstyle{1\over 2}}u)\sin v\cos v.$$ (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

$$x(u,v) = [a+\cos({\textstyle{1\over 2}}u)\sin(v)-\sin({\textstyle{1\over 2}}u)\sin(2v)]\cos(u)$$ (5)

$$y(u,v) = [a+\cos({\textstyle{1\over 2}}u)\sin(v)-\sin({\textstyle{1\over 2}}u)\sin(2v)]\sin(u)$$ (6)

$$z(u,v) = \sin({\textstyle{1\over 2}}u)\sin(v)+\cos({\textstyle{1\over 2}}u)\sin(2v)$$ (7)

for $u\in [0,2\pi)$, $v\in[0,2\pi)$, and $a>2$ (Gray 1993).

The image of the Cross-Cap map of a Torus centered at the Origin is a Klein bottle (Gray 1993, p. 249).

Any set of regions on the Klein bottle can be colored using six colors only (Franklin 1934, Saaty and Kainen 1986).

See also Cross-Cap, Etruscan Venus Surface, Ida Surface, Map Coloring, Möbius Strip

References

Dickson, S. ``Klein Bottle Graphic.'' http://www.mathsource.com/cgi-bin/MathSource/Applications/Graphics/3D/0201-801.

Franklin, P. ``A Six Colour Problem.'' J. Math. Phys. 13, 363-369, 1934.

Geometry Center. ``The Klein Bottle.'' http://www.geom.umn.edu/zoo/toptype/klein/.

Geometry Center. ``The Klein Bottle in Four-Space.''
http://www.geom.umn.edu/~banchoff/Klein4D/Klein4D.html.

Gray, A. ``The Klein Bottle.'' §12.4 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 239-240, 1993.

Nordstrand, T. ``The Famed Klein Bottle.'' http://www.uib.no/people/nfytn/kleintxt.htm.

Pappas, T. ``The Moebius Strip & the Klein Bottle.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 44-46, 1989.

Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 45, 1986.

Stewart, I. Game, Set and Math. New York: Viking Penguin, 1991.

Wang, P. ``Renderings.'' http://www.ugcs.caltech.edu/~peterw/portfolio/renderings/.