Image:Discriminant real part.jpeg

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Discriminant_real_part.jpeg‎ (67KB, MIME type: image/jpeg)

Modular discrimnant, real part, as function of nome. (600x600 pixels)

1 Detailed description
2 Source of Image
3 Copyright status
4 Relevant Links

[edit] Detailed description

This image shows the real part of the modular discriminant $\Delta=g_2^3-27g_3^2=(2\pi)^{12}\eta^{24}$ as a function of the square of the nome $q = exp(i πτ)$ on the unit disk |q| < 1. That is, $πτ$ runs from 0 to $2π$ along the edge of the disk. Black areas indicate regions where the real part is zero or negative; blue/green areas where the value is small and positive, yellow/red where it is large and positive. 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 the above, g₂=60 G₄ and g₃=140 G₆ are Weierstrass elliptic functions invarients, whereas $η$ is the Dedekind eta function.

See also Image:Gee_three_real.jpeg and Image:Gee_three_imag.jpeg for the real and imaginary parts of g₃.

The imaginary part is quite similar. 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 14 February 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

This picture/multimedia file is now available on Wikimedia Commons as Image:Discriminant real part.jpeg.
Images which have been tagged with this template may be deleted immediately after satisfying these conditions (CSD I9).

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