A four-state automaton turns each indexed ± sign rule into a striking self-similar walk. Explore compact, branching, woven, and reflective forms.
Explore the rule family →The first 500,001 vertices of the proved Gaussian-lattice walk. Drag to orbit, scroll to zoom, and right-drag or Shift-drag to pan.
The planar coordinates encode the Gaussian direction in their low bits. The height uses the same exponent modulo four. Every exact step satisfies |dx|, |dy| ≤ 2 and 1 ≤ dz ≤ 7.
The height control changes only the on-screen aspect ratio. It never changes the stored integer coordinates or the proof artifact.
The theorem uses the containing box \([-3,3]^2\times[1,7]\). This prefix realizes exactly the vectors drawn below; the white point is their common origin. Drag the diagram to rotate it.
Each colored dot is \(w_n=2z_n+c_{\alpha_n}\). Its color shows the direction \(i^{\alpha_n}\). Click any two dots to verify \(\nu_2(|w_n-w_m|^2)=\nu_2(h_n-h_m)\).