Breather surface – a pseudosphere like a flower

Inspired by Juan Carlos on X:

#mathart #parametric #surfaces https://t.co/kY3gJxHvd1 pic.twitter.com/GbZ16Yyw5U

— ∞ 𝕁uan ℂarlos (@jcponcemath) April 21, 2025

A Breather surface is beautiful, a pseudosphere famous in differential geometry and theoretical physics. A parametrization is here.

I plotted it with pgfplots and TikZ in LaTeX, and colored it like a flower, using the colormaps feature. A gradient background enhances the appearance.

\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.8}
\pgfplotsset{trig format plots=rad}
\usetikzlibrary{backgrounds}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
    view = {60}{-60},
    hide axis,
    colormap = {flower}{%
      color(0cm)  = (yellow!20);
      color(4cm)  = (orange!50!yellow);
      color(8cm)  = (red!40!black);
      color(12cm) = (red!80!black); 
      color(16cm) = (green);
      color(20cm) = (green!20!black)},
    /tikz/background rectangle/.style = {
        left color    = blue!60,
        right color   = blue!10,
        shading angle = 135},
    show background rectangle]
  \addplot3[surf,
    z buffer = sort,       point meta = u,
    domain   = -13.2:13.2, domain y   = -37.4:37.4,
    samples  = 30, samples y  = 30,% take 80 and 120 for the image below
    variable = \u, variable y = \v ] 
  ( { -u + (2*0.84*cosh(0.4*u)*sinh(0.4*u))/
    (0.4*((sqrt(0.84)*cosh(0.4*u))^2
      + (0.4*sin(sqrt(0.84)*v))^2)) },
    { (2*sqrt(0.84)*cosh(0.4*u)*(-(sqrt(0.84)*sin(v)
      * cos(sqrt(0.84)*v)) + cos(v)*sin(sqrt(0.84)*v)))/(0.4
      * ((sqrt(0.84)*cosh(0.4*u))^2 + (0.4*sin(sqrt(0.84)*v))^2)) },
    { (2*sqrt(0.84)*cosh(0.4*u)*(-(sqrt(0.84)*cos(v)
      * cos(sqrt(0.84)*v)) - sin(v)*sin(sqrt(0.84)*v)))/(0.4
      * ((sqrt(0.84)*cosh(0.4*u))^2 + (0.4*sin(sqrt(0.84)*v))^2)) });
\end{axis}
\end{tikzpicture}
\end{document}