Detailed description

This image shows the phase ${\displaystyle \arctan(\Im j/\Re j)}$ of the J-invariant ${\displaystyle j=g_{2}^{3}/\Delta }$ as a function of the square of the nome ${\displaystyle q=\exp(i\pi \tau )}$ on the unit disk ${\displaystyle |q|<1}$. That is, ${\displaystyle \pi \tau }$ runs from 0 to ${\displaystyle 2\pi }$ along the edge of the disk. Black indicates regions where the phase is ${\displaystyle -\pi }$, green where the phase is zero, and red where the phase is ${\displaystyle +\pi }$. Zeros occur at the points where the colors wrap all the way around a point. Here, the tri-corner intersections show that the phase wraps around three times, for a total of ${\displaystyle 6\pi }$, indicating that the zeros are of the third power: ${\displaystyle (q-q_{0})^{3}}$.

The diamond-shaped patterns at the right side of the image are Moiré patterns, and are an artifact of the pixelization of the image (the 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. In this sense, this particular image clearly illustrates the tesselation of the Poincare disk by the modular group. Each quadrilateral visible in the image consists of a pair of hyperbolic triangles; each triangle is a fundamental domain of the modular group. Note in particular that one corner of each triangle lies on the edge of the disk, with exactly one exception: there is one exceptional very tiny triangle (about two pixels in size), taking the shape of an oval, that lies surrounding the center of the disk. One corner of that triangle is exactly at the center ${\displaystyle q=0}$. See the image of the real part for a description of this exceptional triangle, as well as the funny exceptional tongue that goes with it.

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

Relevant Links

• Weierstrass elliptic functions
• Eisenstein series
• Q-series