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f(x,y) = (((~(11 * y)) ^ ((~x) * (x + x))) & (~((x - 11) | (y + x)))) % 6

Extent: 256x256 (scaled x2)

"Onebit" colouring scheme.

My Emacs pinky is flaring up again, so trying to use the right Ctrl key more. Muscle memory is dang hard to overcome!

This comes from the matrix
$$
L = \begin{pmatrix}
V(1) & 1 & 0 & \cdots & 0 & 1 \\
1 & V(2) & 1 & & & 0 \\
0 & 1 & V(3) & 1 & & \vdots \\
\vdots & & & \ddots & & 0 \\
0 & & & 1 & V(q-1) & 1 \\
1 & 0 & \cdots & 0 & 1 & V(q)
\end{pmatrix},
$$
where $$V(k) = 2 \cos(2 \pi k p/q)$$ for integer p and q.

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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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