Difference between revisions of "Kummer Quartic"

From Math Images
Jump to: navigation, search
Line 5: Line 5:
 
|ImageIntro=A Kummer surface is any one of a one parameter family of algebraic surfaces defined by a specific polynomial equation of degree four.
 
|ImageIntro=A Kummer surface is any one of a one parameter family of algebraic surfaces defined by a specific polynomial equation of degree four.
 
|ImageDesc=<nowiki>The polynomial equation of degree four that describes a Kummer surfaces is:</nowiki>
 
|ImageDesc=<nowiki>The polynomial equation of degree four that describes a Kummer surfaces is:</nowiki>
<math>(x^2 + y^2 + z^2 - aa^2)^2 - \lambda\*p*q*r*s = 0,</math> where
+
<math>(x^2 + y^2 + z^2 - aa^2)^2 - \Lambda\*p*q*r*s = 0,</math> where
 
<math>aa</math> is any real number,
 
<math>aa</math> is any real number,
<math>\lambda\ = (3*aa^2 - 1.0)/(3 - aa^2)</math>,
+
<math>\Lambda\ = (3*aa^2 - 1.0)/(3 - aa^2)</math>,
 
<math>p = 1 - z - \sqrt\{2}*x</math>,
 
<math>p = 1 - z - \sqrt\{2}*x</math>,
 
<math>q = 1 - z + \sqrt\{2}*x</math>,
 
<math>q = 1 - z + \sqrt\{2}*x</math>,

Revision as of 16:15, 16 July 2008


Kummer Quartic
Kummer.png
Field: Algebra
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

The polynomial equation of degree four that describes a Kummer surfaces is: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (x^2 + y^2 + z^2 - aa^2)^2 - \Lambda\*p*q*r*s = 0, where

'"`UNIQ--math-00000002-QIN [...]

The polynomial equation of degree four that describes a Kummer surfaces is: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (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), Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): p = 1 - z - \sqrt\{2}*x , Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): q = 1 - z + \sqrt\{2}*x , Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): r = 1 + z + \sqrt\{2}*y , Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 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.




Teaching Materials (1)

Teaching Materials (1)

Want to add your own teaching materials or lesson plans? Add teaching materials.

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.








If you are able, please consider adding to or editing this page!


Have questions about the image or the explanations on this page?
Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.