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# Kummer Surface

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

 (1)

where

 (2)

, , , and are the tetrahedral coordinates

 (3) (4) (5) (6)

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)

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.

Kummer Surface

## Cite this as:

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