Question 1

What is the Hopf fibration?

Accepted Answer

The Hopf fibration is a way of mapping a 4-dimensional sphere (S³) onto an ordinary 3D sphere (S²). Every point on the 3D sphere corresponds to a circle in 4D space — those circles are the 'fibers'. When projected back into our 3D world, they form interlocking rings that fill space without ever touching. Discovered by Heinz Hopf in 1931.

Question 2

What does Hopf Skip do?

Accepted Answer

Hopf Skip is an interactive playground for the Hopf fibration. It generates the fibers in 4D, applies stereographic projection to bring them into 3D, and renders them in your browser with WebGL. You can pick from 12 topology layouts, change colors with generative palettes, switch between tube and ribbon fibers, and export 4K images.

Question 3

What are the 12 topology presets?

Accepted Answer

Keyring (single equator ring), Nested (concentric tori shells), Uniform (Fibonacci-distributed), Meridian (pole-to-pole arc), Spiral (helical sweep), Orthogonal (perpendicular rings), Lissajous (woven harmonic), Wave (oscillating equator), Rose (rose-curve modulated), Helix (twin helices), Cage (lat-long basket), and Crown (alternating saw-tooth). Each is a different sampling pattern on the base sphere S².

Question 4

What are Villarceau circles and the Clifford torus?

Accepted Answer

When the Hopf fibers are stereographically projected, the rings over a circle of latitude on S² form a torus made of linking Villarceau circles — the diagonal slices of a torus. The flat torus at the equator is the Clifford torus, the most symmetric case. Every two fibers from different points are linked exactly once but never intersect.

Question 5

How do I use Hopf Skip?

Accepted Answer

Tap the canvas (or press Space) to evolve a new random configuration. Drag to orbit, pinch or scroll to zoom. Open the menu (top-right) to tune topology, density, color harmony, material, and 4D twist. Press R to reset, E to render a 4K PNG, ? for this info.

Question 6

Can I save and share my creation?

Accepted Answer

Yes. Save State exports a JSON file with the full configuration AND copies a shareable hash-link to your clipboard. Load State imports a JSON file. Render 4K saves a 3840×2160 PNG suitable for wallpapers, prints, or social posts. Nothing leaves your device unless you choose to share it.

Question 7

Why does the rendering sometimes slow down?

Accepted Answer

Each fiber is full 3D geometry. At maximum density (200+ fibers, nested mode with 12 layers), you're rendering several thousand rings. On older mobile devices, drop fibers below 100 or pick a single-shell topology like Keyring or Spiral. Hopf Skip clamps density automatically to keep things stable.

Question 8

Is Hopf Skip free? Does it collect my data?

Accepted Answer

Free, forever. Hopf Skip runs entirely in your browser with no account, no signup, and no data collection beyond anonymous Google Analytics. Your saved states are stored locally on your device only.

HOPF SKIP

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Hopf Skip — Interactive Hopf Fibration & 4D Topology Playground