“Klein Bottle”的概念、定义、习题解题方法及翻译-数学辞典

单词

Klein Bottle

释义

Klein Bottle

A closed Nonorientable Surface of Genus one having no inside or outside. It can bephysically realized only in 4-D (since it must pass through itself without the presence of a Hole). ItsTopology is equivalent to a pair of Cross-Caps with coinciding boundaries. It can be cut in halfalong its length to make two Möbius Strips.

The above picture is an Immersion of the Klein bottle in (3-space). There is also another possibleImmersion called the ``figure-8'' Immersion (Geometry Center).

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 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/.

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