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Kummer Quartic
Fields: Algebra and Geometry
Image Created By: 3DXM Consortium
Website: Virtual Math Museum

Kummer Quartic

A Kummer surface is any one of a one parameter family of algebraic surfaces defined by a specific polynomial equation of degree four.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Differential Geometry, Algebra

The polynomial equation of degree four that describes a Kummer surfaces is:

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The polynomial equation of degree four that describes a Kummer surfaces is: $(x^2 + y^2 + z^2 - aa^2)$$^2$$- \lambda *p*q*r*s = 0$, where:

$aa$ is any real number,
$\lambda\ = (3*aa^2 - 1.0)/(3 - aa^2)$,
$p = 1 - z - \sqrt{2}*x$,
$q = 1 - z + \sqrt{2}*x$,
$r = 1 + z + \sqrt{2}*y$,
and $s = 1 + z - \sqrt{2}*y$.

The family was described originally by Ernst Eduard Kummer in 1864.

A Kummer surface has sixteen double points, the maximum possible for a surface of degree four in three-dimensional space. For the default case $aa$ = 1.3, all these double points are real and they appear in the visualization as the vertices of five tetrahedra.

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About the Creator of this Image

The 3DXM Consortium is the group in charge of the 3D-XplorMath software development project and the related Virtual Mathematics Museum website project. The Consortium is an international volunteer group of mathematicians.

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