The Kummer surfaces are a family of quartic surfaces
given by the algebraic equation
(1)
where
(2)
, , ,
and
are the tetrahedral coordinates
and
is a parameter which, in the above plots, is set to .
The above plots correspond to
(7)
(double sphere ), 2/3, 1
(8)
(Roman surface ), 2, 3
(9)
(four planes), and 5. The case corresponds to four real points.
The following table gives the number of ordinary double points for various ranges of , corresponding to the preceding illustrations.
parameter real nodes complex nodes 4 12 4 12 16 0 16 0
The Kummer surfaces can be represented parametrically by hyperelliptic theta functions. Most of the Kummer surfaces admit 16 ordinary
double points , the maximum possible for a quartic
surface . A special case of a Kummer surface is the tetrahedroid .
Nordstrand gives the implicit equations as
(10)
or
(11)
See also Desmic Surface ,
Quartic
Surface ,
Roman Surface ,
Tetrahedroid
Explore with Wolfram|Alpha
References Endraß, S. "Flächen mit vielen Doppelpunkten." DMV-Mitteilungen 4 , 17-20, Apr. 1995. Endraß, S.
"Kummer Surfaces." http://enriques.mathematik.uni-mainz.de/docs/Ekummer.shtml . Fischer,
G. (Ed.). Mathematische
Modelle aus den Sammlungen von Universitäten und Museen, Kommentarband.
Braunschweig, Germany: Vieweg, pp. 14-19, 1986. Fischer, G. (Ed.).
Plates 34-37 in Mathematische
Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig,
Germany: Vieweg, pp. 33-37, 1986. Gray, A. Modern
Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca
Raton, FL: CRC Press, p. 313, 1997. Guy, R. K. Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 183,
1994. Hudson, R. W. H. T. Kummer's
Quartic Surface. Cambridge, England: University Press, 1905. Reprinted Cambridge,
England: Cambridge University Press, 1990. Kummer, E. "Über
die Flächen vierten Grades mit sechszehn singulären Punkten." Collected
Papers, Volume 2: Functions, Theory, Geometry and Miscellaneous (Ed. A. Weil).
Berlin: Springer-Verlag, pp. 418-432, 1975. Kummer, E. "Über
Strahlensysteme, deren Brennflächen Flächen vierten Grades mit sechszehn
singulären Punkten sind." Collected Papers, Volume 2: Functions, Theory,
Geometry and Miscellaneous (Ed. A. Weil). Berlin: Springer-Verlag,
pp. 418-432, 1975. Nordstrand, T. "Kummer's Surface."
http://jalape.no/math/kummtxt . Referenced
on Wolfram|Alpha Kummer Surface
Cite this as:
Weisstein, Eric W. "Kummer Surface." From
MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/KummerSurface.html
Subject classifications