Vector projection

Klein's J-invariant, real part (600x600 pixels)

Detailed description

This image shows the real part $\Re j$ of the J-invariant $j=g_{2}^{3}/\Delta$ as a function of the square of the nome $q=\exp(i\pi \tau )$ on the unit disk |q| < 1. That is, $\pi \tau$ runs from 0 to $2\pi$ 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 diamond-shaped patterns in the red part are Moiré patterns, and are an artifact of the pixelization of the image (the red-black strips are smaller than the size of a pixel; the color of the pixel is assigned according to the value of the function at the center of the pixel, rather than the average of values over the pixel).

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.

The imaginary part $\Im j$ looks roughly similar; the modulus $|j|={\sqrt {(\Re {j})^{2}+(\Im {j})^{2}}}$ is uniform in color, with the black strips filled in to match the colored areas. The modulus essentially looks like Image:Q-euler.jpeg with the colors reversed.

Zeros are visible where-ever the three-pointed triangle tips come together. The j-invariant has a pole at every rational multiple of π on the circumference of the disk. The correct way to understand this image is to note that j takes on every possible value on the fundamental region. Each fundamental region takes the form of a hyperbolic triangle in this image, with one vertex of the triangle on the edge of the disk. Thus, the red regions deceivingly hint that they are centered on a pole; this is not the case, as the poles lie on the disk boundary. There is exactly one exception to this: there is one very tiny triangle (about two pixels in size), taking the shape of an oval, that is wrapped around the dead-center of the disk. One corner of that triangle is exactly at the center q=0, with a pair of edges zipped together running from q=0 to $q=-\exp(-\pi {\sqrt {3}})$ (which is about -0.0043, which is why its not visible here). The third edge of this exceptional triangle circles the origin all the way around. This third edge is shared with the unique funny-looking tongue in this image, turning this seemingly two-sided tongue into a real triangle. Note that this implies that the j-function has a simple pole at the origin, although it is not visible in this image.

See also Image:J-inv-phase.jpeg for the phase part. It, and other related images, can be seen at http://www.linas.org/art-gallery/numberetic/numberetic.html

Source of Image

Created by Linas Vepstas User:Linas <linas@linas.org> on 15 February 2005 using custom software written entirely by Linas Vepstas.

Copyright status

Released under the Gnu Free Documentation License (GFDL) by Linas Vepstas. Template:GFDL-with-disclaimers

Relevant Links

• Weierstrass elliptic functions
• Eisenstein series
• Q-series

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