Image:J-inv-poincare.jpeg

From Wikipedia, the free encyclopedia

Image
File history
File links

No higher resolution available.
J-inv-poincare.jpeg‎ (600 × 600 pixels, file size: 37 KB, MIME type: image/jpeg)

Klein's J-invariant, modulus on Poincaré disk (600x600 pixels)

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

[edit] Detailed description

This image shows the modulus $| j |$ of the J-invariant $j=g_2^3/\Delta$ as a function on the Poincare disk. The Poincaré disk is a mapping of the upper half-plane to the unit disk by means of the Mobius transformation

$z=\frac{\tau-i}{\tau+i}$

where τ is the half-period ratio, that is, the coordinate on the upper half-plane. The left-right symmetry of this image corresponds to the $\tau\to -1/\tau$ symmetry of the J-invariant, as small values for the imaginary part of τ are mapped to z=−1 on the left hand side, while large values for the imaginary part of τ are mapped to z=+1 on the right side. The fundamental domain is a triangle spanning the two vertically oriented dots in the center, with the cusp at z=+1 on the right. The transformation $\tau\to\tau+1$ corresponds to walking the fundamental domain from dot to dot on the boundary of the largest red area on the right; the cusp z=+1 remains unchanged.

For a detailed description of other aspects of this picture, including the coloration, please refer to Image:J-inv-modulus.jpeg.

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

Click on a date/time to view the file as it appeared at that time.

	Date/Time	Dimensions	User	Comment
current	01:30, 22 May 2005	600×600 (37 KB)	Linas (Talk \| contribs)	(Klein's J-invariant, modulus on Poincare disk (600x600 pixels) )