If you're into chaos, you might enjoy this visualization of the standard map
\begin{align}
p_{n+1} = p_n + K \sin \theta_n, \\
\theta_{n+1} = \theta_n + p_{n+1},
\end{align}
with varying $$K$$. It's pretty hypnotic shadertoy.com/view/WdjcW3

As it turns out, there is a better approach using Clenshaw's algorithm! A review of different methods is given in arXiv:1410.1748 [physics.chem-ph]. Now I could get up to high values of $$\ell$$ and $$m$$ without everything blowing up! From the ALPs we can then simply obtain the SHs as
$Y_\ell^m(\theta, \phi) = \sqrt{\frac{(2 \ell + 1) (\ell - m)!}{4\pi (\ell + m)!}} P_\ell^m (\cos \theta) e^{i m \phi}.$
The final code can be viewed at shadertoy.com/view/3dKGWG, which borrows from iq's shader

Maybe I should do a small

Hello everyone! I'm Ron(i) and go by Roninkoi in most places. As a kid I started off making games for competitions such as , but these days I'm more into the technical side of things. Graphics programming and is my main interest, especially and . IRL I'm a computational grad student working as a part-time code monkey

Hofstadter's butterfly is a fractal arising from non-interacting electrons in a magnetic field. We can draw it by computing the eigenvalues of the system's Hamiltonian and plotting values of the Lyapunov exponent. Here's a shader that does just that! shadertoy.com/view/7s2XRt

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