---
title: "Why is this hyperbolic background invariant?"
author: "Stéphane Laurent"
date: '2023-11-20'
tags: R, graphics, maths, special-functions
rbloggers: yes
output:
  md_document:
    variant: markdown
    preserve_yaml: true
  html_document:
    highlight: kate
    keep_md: no
highlighter: pandoc-solarized
---

I have a mathematical question for you, because it is a mystery for me.

Look at this animation:

![](./figures/lambdaOnSquare.gif)

I get it as follows. Each frame corresponds to a value of $t \in [0,3[$
(I take $160$ values of $t$ subdivising $[0,3[$). Here is how I get the
frame corresponding to one value of $t$:

-   for each point in the unit square $S = {[0,1]}^2$, I take its
    complex affix $z$, and I send $z$ to the open upper half-plane
    $\mathbb{H} = \bigl\{z | \Im(z) > 0\bigr\}$ with a conformal map
    $\psi$ from $S$ to $\mathbb{H}$;

-   I attribute a color to $R^t\Bigl(\lambda\bigl(\psi(z)\bigr)\Bigr)$
    where $R$ is the Möbius transformation of order $3$ defined by
    $R(z) = -\dfrac{1}{z+1}$ and $\lambda$ is the modular lambda
    function.

The modular lambda function and the conformal map $\psi$ are implemented
in my R package **jacobi**.

My question is: why does the hyperbolic tessellation that we can see as
the "background" of the animation not move? Why is it invariant?

I observed the same phenomenon for other modular functions, for example
the Klein j-invariant function.

The color mapping $\mathcal{C}$ is defined with the help of the HSI
color space. The color $\mathcal{C}(z)$ depends on the phase of $z$
only. Precisely, $\mathcal{C}(z)$ is the HSI color with:

-   *hue* $\in [0, 360[$ given by the phase $\varphi(z) \in [0, 2\pi[$
    of $z$ converted to degrees;

-   *saturation* given by $\sqrt{\bigl(1 + \sin(w)\bigr) / 2}$ where
    $w = 2\pi\log\bigl(1+\varphi(z)\bigr)$;

-   *intensity* given by $\bigl(1 + \cos(w)\bigr) / 2$ with $w$ as
    above.

Why do we get such a result? I really don't know.

Before leaving you, let me show you a 3D version of this animation that
I made with the
[**isocuboids**](https://github.com/cj-holmes/isocuboids) R package:

![](./figures/isocuboids_lambdaOnSquare-gray.gif)

I hope you like it.