Image:Discriminant real part.jpeg

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[edit] Summary

originally uploaded to en:wiki, with this history:

  • 04:11, 15 February 2005 . . Linas (Talk | contribs) . . 600×600 (67,813 bytes) (Modular discrimnant, real part, as function of nome.)

Comments made by the author follow.

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

[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(iπτ) 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.

[edit] Licensing

GNU head 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.

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