Difference between revisions of "Kummer Quartic"

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A Kummer surface has sixteen double points, the maximum possible for a surface of degree four in three-dimensional space. For the default case <math>aa</math> = 1.3, all these double points are real and they appear in the visualization as the vertices of five tetrahedra.
 
A Kummer surface has sixteen double points, the maximum possible for a surface of degree four in three-dimensional space. For the default case <math>aa</math> = 1.3, all these double points are real and they appear in the visualization as the vertices of five tetrahedra.
|s1a=No
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|other=Differential Geometry, Algebra
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|AuthorName=3DXM Consortium
 
|AuthorName=3DXM Consortium
 
|AuthorDesc=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.
 
|AuthorDesc=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.
 
|SiteName=Virtual Math Museum
 
|SiteName=Virtual Math Museum
 
|SiteURL=http://virtualmathmuseum.org/
 
|SiteURL=http://virtualmathmuseum.org/
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|TeachingMaterials=:* A teaching material test. [[Kummer Quartic Teaching]]
 
|Field=Algebra
 
|Field=Algebra
 
|Field2=Geometry
 
|Field2=Geometry
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|FieldLinks=:* [http://xahlee.org/surface/kummer/_jv_kummer.html Rotate a Kummer Quartic]
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:* [http://mathworld.wolfram.com/KummerSurface.html Kummer Surfaces - Wolfram Mathworld]
 
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<accesscontrol>Math-Images-Staff</accesscontrol>

Latest revision as of 15:09, 15 January 2010


Kummer Quartic
Kummer.png
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:

'"`UNIQ--math-00000000-Q [...]

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.




Teaching Materials (1)

Teaching Materials (1)

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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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<accesscontrol>Math-Images-Staff</accesscontrol>