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Kummer Surface

\begin{figure}\begin{center}\BoxedEPSF{KummerSurface.epsf scaled 1500}\end{center}\end{figure}

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

(x^2+y^2+z^2-\mu^2w^2)^2-\lambda pqrs=0,
\end{displaymath} (1)

\lambda\equiv {3\mu^2-1\over 3-\mu^2},
\end{displaymath} (2)

$p$, $q$, $r$, and $s$ are the Tetrahedral Coordinates
$\displaystyle p$ $\textstyle =$ $\displaystyle w-z-\sqrt{2}\,x$ (3)
$\displaystyle q$ $\textstyle =$ $\displaystyle w-z+\sqrt{2}\,x$ (4)
$\displaystyle r$ $\textstyle =$ $\displaystyle w+z+\sqrt{2}\,y$ (5)
$\displaystyle s$ $\textstyle =$ $\displaystyle w+z-\sqrt{2}\,y,$ (6)

and $w$ is a parameter which, in the above plots, is set to $w=1$. The above plots correspond to $\mu^2=1/3$
\end{displaymath} (7)

(double sphere), 2/3, 1
\end{displaymath} (8)

(Roman Surface), $\sqrt{2}$, $\sqrt{3}$
\end{displaymath} (9)

(four planes), 2, and 5. The case $0\leq\mu^2\leq 1/3$ 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.

$0\leq\mu^2\leq {\textstyle{1\over 3}}$ 4 12
$\mu^2={\textstyle{1\over 3}}$    
${\textstyle{1\over 3}}\leq\mu^2<1$ 4 12
$1<\mu^2<3$ 16 0
$\mu^2>3$ 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

x^4+y^4+z^4-x^2-y^2-z^2-x^2y^2-x^2z^2-y^2z^2+1 = 0
\end{displaymath} (10)


\end{displaymath} (11)

See also Quartic Surface, Roman Surface, Tetrahedroid


Endraß, S. ``Flächen mit vielen Doppelpunkten.'' DMV-Mitteilungen 4, 17-20, Apr. 1995.

Endraß, S. ``Kummer Surfaces.''

Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp. 14-19, 1986.

Fischer, G. (Ed.). Plates 34-37 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 33-37, 1986.

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 183, 1994.

Hudson, R. Kummer's Quartic Surface. Cambridge, England: Cambridge University Press, 1990.

Kummer, E. ``Über die Flächen vierten Grades mit sechszehn singulären Punkten.'' Ges. Werke 2, 418-432.

Kummer, E. ``Über Strahlensysteme, deren Brennflächen Flächen vierten Grades mit sechszehn singulären Punkten sind.'' Ges. Werke 2, 418-432.

Nordstrand, T. ``Kummer's Surface.''

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© 1996-9 Eric W. Weisstein