Resumo

originally uploaded to en:wiki, with this history:

• 04:01, 15 February 2005 . . Linas (Talk | contribs) . . 600×600 (52,911 bytes) (image changed to show square of nome instead)
• 05:43, 14 February 2005 . . Linas (Talk | contribs) . . 600×600 (53,758 bytes) (Weierstrass invarient g three.jpeg)

Comments made by the author follow.

Weierstrass elliptic functions invariant g₃, real part (600x600 pixels)

Detailed description

This image shows the real part of the Weierstrass elliptic functions (wp) invariant g₃=140 G₆ 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; 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.

See also Image:Gee_three_imag.jpeg for the imaginary part; both images use the same color scale, and thus give a hint of the relative magnitudes.

The corresponding image for g₂ is quite similar. It, and other related images, can be seen at http://www.linas.org/art-gallery/numberetic/numberetic.html

The images for general Eisenstein series G_2n will appear to be similar, but more intensely filigreed. In particular, notice how this image has two colored lobes running down the middle. The corresponding image for G_2n will show n-=1 lobes.

Source of Image

Created by Linas Vepstas User:Linas <linas@linas.org> on 13 February 2005 using custom software written entirely by Linas Vepstas.

Copyright status

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

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