Modular Form Level 1, Weight 12 (1.12.a.a) Visualised
Modular forms are highly self-symmetric functions studied in number theory. They have surprising connections to several different fields of mathematics. Modular forms are rarely visualised, especially not in 3D. David Lowry-Duda reached out to me after seeing some of my other graphics on YouTube and we set to work attempting to create some new visualisations. These visualisations were first shared by David at the 2021 Bridges conference. (Paper). (Slides)
Thanks to David Lowry-Duda for his work coding the Modular Form calculation. I've used his procedure to calculate only on the fundamental domain, which improves accuracy.
Click on each image to view the high-quality 8k version. The images on this page are licensed under the Creative Commons Attribution 4.0. You may use them in your own work if you give credit. Please credit "Maths Town and David Lowry-Duda", where appropriate please link to this page.
This form is often known as the "Modular Discriminant" and the coefficients of the q-series are the Ramanujan tau function.
Due to the very large values, we have applied a damping function to control the height of the 3D outputs.
The colour represents the argument (phase) of the function. Where hue=arg(z). Red is the positive real axis, cyan is the negative real axis.
Poincaré disk
The Poincaré disk is a conformal mapping of the upper-half plane onto the unit circle.
Height damping function: atan(log(sqrt(z)+1)), then scaled for visual appeal.
Modular Form Weight 12. (Poincaré disk plot 1) [CC BY 4.0]
Modular Form Weight 12. (Poincaré disk plot 2) [CC BY 4.0]
Modular Form Weight 12. (Poincaré disk plot 3) [CC BY 4.0]