File:Discriminant real part.jpeg

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• 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 discriminant, real part, as function of nome. (600x600 pixels)

This image shows the real part of the modular discriminant ${\displaystyle \Delta =g_{2}^{3}-27g_{3}^{2}=(2\pi )^{12}\eta ^{24}}$ as a function of the square of the nome ${\displaystyle q=\exp(i\pi \tau )}$ on the unit disk |q| < 1. That is, ${\displaystyle \pi \tau }$ runs from 0 to ${\displaystyle 2\pi }$ 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 ${\displaystyle \eta }$ 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

Created by

Linas Vepstas
Alternative names	L. Vepstas
Description	technologist, entrepreneur, researcher, programmer and computer scientist GnuCash maintainer for over 7 years.
Authority file	: Q61131517 VIAF: 311397845 ORCID: 0000-0002-2557-740X

on 14 February 2005 using custom software written entirely by Linas Vepstas.

Released under the Gnu Free Documentation License (GFDL) by Linas Vepstas.

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