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Kummer Surface

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KummerSurface

The Kummer surfaces are a family of quartic surfaces given by the algebraic equation

(x^2+y^2+z^2-mu^2w^2)^2-lambdapqrs=0,
(1)

where

lambda=(3mu^2-1)/(3-mu^2),
(2)

p, q, r, and s are the tetrahedral coordinates

p=w-z-sqrt(2)x
(3)
q=w-z+sqrt(2)x
(4)
r=w+z+sqrt(2)y
(5)
s=w+z-sqrt(2)y,
(6)

and w is a parameter which, in the above plots, is set to w=1.

The above plots correspond to mu^2=1/3

(3x^2+3y^2+3z^2-1)^2=0,
(7)

(double sphere), 2/3, 1

x^4-2x^2y^2+y^4+4x^2z+4y^2z+4x^2z^2+4y^2z^2=0
(8)

(Roman surface), 2, 3

[(z-1)^2-2x^2][2y^2-(z+1)^2]=0
(9)

(four planes), and 5. The case 0<=mu^2<=1/3 corresponds to four real points.

The following table gives the number of ordinary double points for various ranges of mu^2, corresponding to the preceding illustrations.

parameterreal nodescomplex nodes
0<=mu^2<=1/3412
mu^2=1/3
1/3<=mu^2<1412
mu^2=1
1<mu^2<3160
mu^2=3
mu^2>3160

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

x^4+y^4+z^4-x^2-y^2-z^2-x^2y^2-x^2z^2-y^2z^2+1=0
(10)

or

x^4+y^4+z^4+a(x^2+y^2+z^2)+b(x^2y^2+x^2z^2+y^2z^2)+cxyz-1=0.
(11)

See also

Desmic Surface, Quartic Surface, Roman Surface, Tetrahedroid

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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.

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Kummer Surface

Cite this as:

Weisstein, Eric W. "Kummer Surface." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KummerSurface.html

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