Shiftless Collatz Billiards 🎱

Author: Hiroshi Harada
Date: March 27, 2026
License: Code under MIT, Generated Media under CC BY 4.0

Overview

This repository provides Python visualization tools for exploring the Shiftless Collatz Model, a rewrite system that is mathematically equivalent to the classical Collatz iteration but preserves all binary information.

Unlike the standard $3n+1$ process, which repeatedly discards information through right-shifts (division by 2), the Shiftless Collatz Model evolves according to:

$$n_{k+1} = 3n_k + 2^{v_2(n_k)}$$

This formulation reveals a hidden geometric structure when plotted on a $\log_3$ lattice plane.
The trajectory behaves like a billiard ball driven by:

• Inertia from the $3n$ multiplier
• Warp from the LSB interference term $2^{v_2(n_k)}$
• Jackpot pockets located at the powers of two, $2^M$

The result is a visually striking dynamical system where the orbit bends, oscillates, and ultimately converges into a $2^M$ “jackpot line.”

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Files Included

collatz_billiards_image.py

Generates a high-resolution (300 DPI) static overview of the entire Shiftless Collatz trajectory.
Ideal for research papers, posters, and static figures.

collatz_billiards_animation.py

Exports a cinematic tracking-zoom animation (GIF and optionally MP4) featuring:

• Tile-by-tile camera movement
• Pre- and post-transition pauses
• Jackpot zoom-in moments
• A final overview freeze frame

Perfect for presentations, lectures, and visual demonstrations.

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Requirements

pip install numpy matplotlib tqdm

Note: MP4 export requires ffmpeg installed on your system. GIF export works without ffmpeg.

Usage

Run the scripts directly from your terminal or Jupyter environment. You can modify the seed_to_hunt variable at the bottom of each script to explore different trajectories.

# Generate the high-resolution still image
python collatz_billiards_image.py

# Export the animation (GIF and optionally MP4)
python collatz_billiards_animation.py

Example Output (Seed = 27)

• 41-step journey
• Final jackpot at $2^{64}$
• Strong warp oscillations in the early phase
• Multiple near-misses in the mid-phase
• A dramatic final surge with local growth rate up to 3.2

The included GIF demonstrates this behavior.

License

• Python Source Code: MIT License
• Generated Images / Animations: CC BY 4.0
• © 2026 Hiroshi Harada