Image:J-inv-modulus.jpeg
From Wikipedia, the free encyclopedia
J-inv-modulus.jpeg (36KB, MIME type: image/jpeg)
Klein's J-invariant, modulus portrait (600x600 pixels)
Contents |
[edit] Detailed description
This image shows the modulus | j | of the J-invariant 
as a function of the square of the nome q = exp(iπτ) on the unit disk |q| < 1. That is, πτ runs from 0 to 2π along the edge of the disk. Black indicates regions where the modulus is near zero, green where the modulus is about one, and red where the modulus is greater than ten. The color scale is logarithmic. Inside of every black region is a zero; note the zeros are cubic.
The small black dot on the far right, in the large red cardioid, is a numerical artifact.
The fractal self-similarity of this function is that of the modular group; note that this function is a modular form. Every modular function will have this general kind of self-similarity. In this sense, this particular image clearly illustrates the tesselation of the q-disk by the modular group.
- See also Image:J-inv-poincare.jpeg for this J-invariant on the Poincaré disk.
- See also Image:J-inv-real.jpeg for the real part.
- See also Image:J-inv-phase.jpeg for the phase.
It, and other related images, can be seen at http://www.linas.org/art-gallery/numberetic/numberetic.html
[edit] Source of Image
Created by Linas Vepstas User:Linas <linas@linas.org> on 21 May 2005 using custom software written entirely by Linas Vepstas.
[edit] Copyright status
Released under the Gnu Free Documentation License (GFDL) by Linas Vepstas.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
Subject to disclaimers.
[edit] Relevant Links
File history
Legend: (cur) = this is the current file, (del) = delete
this old version, (rev) = revert to this old version.
Click on date to download the file or see the image uploaded on that date.
- (del) (cur) 21:21, 21 May 2005 . . Linas (Talk | contribs) . . 600×600 (36,743 bytes) (Klein's J-invariant, modulus portrait (600x600 pixels))
- Edit this file using an external application
See the setup instructions for more information.
