Kummer Quartic

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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.


Contents

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: (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.




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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