Sturmian Word on 3D Torus

Watch a particle trace a path on a 3D torus: 0 for major circle crossings, 1 for minor circle crossings

Sturmian Sequence

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Unique N-Words

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3D Torus

Unwrapped Torus (Fundamental Domain)

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About 3D Torus Sturmian Words

The particle moves on a 3D torus with parametric equations: θ₁(t) = t and θ₂(t) = slope·t. You can use either an irrational slope (like √2, φ, or π) or a rational ratio p/q.

When the slope is irrational, the path never repeats and creates an aperiodic Sturmian sequence. When it's rational (p/q), the path closes after q loops around the major circle and p loops around the minor circle, creating a periodic sequence.