Visualize dozens of fibers simultaneously, colored by their base point on S²
Hundreds of circles, one for each sampled point on the sphere, woven together so tightly they fill space — and no two of them ever touch.
Step from Sparse to Dense and orbit around the scene. At high density the nested tori become visible. Color marks each fiber's base point: hue for longitude, lightness for latitude.
Now we render many fibers simultaneously, one for each sampled point on S². Each fiber is colored by the position of its base point — longitude determines hue, latitude determines lightness. The result is one of the most beautiful structures in all of mathematics.
Orbit around the scene to appreciate how the circles interweave. Despite being densely packed, no two fibers ever intersect — they fill space without collision.
Increase the density to see more fibers. At high density, the torus-like structure of nested surfaces becomes visible — the 3-sphere is revealed as a continuous family of tori.
Key insight: The rainbow coloring encodes the base point on S². Fibers with similar colors have nearby base points — the fibration is a continuous map.
Drag the latitude slider to watch one torus of fibers morph. Set it to 90° to see the Clifford torus.
Each latitude circle on S² contributes a torus of fibers. Drag the latitude to watch the torus morph. At the equator (θ = 90°) you see the Clifford torus, a flat torus that divides S³ into two congruent solid tori — one of the most remarkable facts in 3-sphere topology.
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