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 (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

(1) |

(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 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).

**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/.

© 1996-9

1999-05-26