| title | Collatz Visualizations |
|---|---|
| emoji | 🧊 |
| colorFrom | blue |
| colorTo | yellow |
| sdk | static |
| pinned | false |
| license | mit |
| short_description | Interactive animations of the 3n+1 problem |
Interactive, self-contained visualizations of the Collatz conjecture (the 3n+1 problem). Everything is plain HTML, CSS and vanilla JavaScript: no build step, no dependencies, and every orbit is computed live in the browser.
Live site: remifabre-collatz-visualizations.static.hf.space
Hailstone Atlas (index.html): seven chapters, from a single animated
orbit to advanced structure.
- The Rule: one seed stepped through the rule on a log₂ altitude chart. Amber segments are odd steps (3n+1), cyan segments are even steps (n/2).
- The Collatz coral: the reverse Collatz tree grown as an organic form (a visual made famous by Edmund Harriss), with tunable branch angles and a tracer that lights up any seed's forward orbit as a path to the root.
- A thousand flights: seeds 2 to 1,000 overlaid, with flight-time record holders highlighted.
- Convergence rain: 2,000 seeds stepping in unison, with a live survival curve (the distribution of total stopping times).
- Stopping-time field: flight length for every seed up to 100,000. The rays are families of seeds whose trajectories merge.
- Spiral traffic: an Ulam spiral where each cell glows by how many trajectories (seeds up to 10,000) pass through it, with a second mode coloring each cell by its own flight length.
- The binary machine: trajectories as bit waterfalls. Halving deletes a bit; 3n+1 is a shift-and-add whose carry ripples are the hard part of the problem.
Hailstone Bestiary (bestiary.html): the same halving dynamics with
the odd step changed. 3n−1 splits the integers between three attracting cycles, 5n+1 and
7n+1 send almost every seed past 2⁴⁵. Includes clickable fate maps and an orbit tracer.
The one-number summary is the drift heuristic: a typical flight gains log₂(q) − 2 bits
per odd step.
Hailstone Music Box (music-box.html): orbits sonified with the
Web Audio API. Pitch follows log₂(n) snapped to C minor pentatonic; odd steps ring
bright. Duet mode plays two seeds in stereo, so you hear the exact step where their
trajectories merge. Seeds use BigInt, so arbitrarily large numbers play exactly.
- Colors are semantic across all pages: warm amber for the 3n+1 rise, cold cyan for the halving fall. The categorical palettes are checked for color-vision-deficiency separation.
- Pages respect
prefers-reduced-motion: they load with completed, static renderings, and any replay/regrow/trace control re-enables animation on demand. - "Escape" in the Bestiary means an orbit was observed passing 2⁴⁵. Divergence is not proven for any single orbit of 5n+1 or 7n+1, just as convergence is not proven for every orbit of 3n+1.
MIT