Base-3 Putter Shot: Friction Dynamics in the Shiftless Collatz Model

“Hunting the truth of the Collatz Conjecture on the base-3 Green”

Author: Hiroshi Harada
License: MIT License
Date: March 29, 2026

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Overview

This repository provides a set of Python visualizers (Static & Animated) for exploring the Collatz conjecture through the lens of the Shiftless model and base-3 polar coordinates.

By interpreting the mathematical transformation as a dynamic putter shot on a logarithmic spiral green, we can observe how the Least Significant Bit (LSB) acts as a physical friction coefficient
$$\mu_k = \frac{\mathrm{LSB}}{B_{k+1}},$$ bending the trajectory step by step until it ultimately falls into a pure power of two ($2^M$).

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Key Features

• Billiard-Style Animation: Step-by-step dynamic rendering that visually represents the “information collisions” occurring at each state.

• Overflow Armor: Safely computes logarithmic values for astronomically large integers without triggering OverflowError, enabling the hunt for massive seeds.

• Scale Locking: Fixes the polar axis limits in advance, ensuring perfectly stable visuals with zero jitter—even during the final flag deployment.

• Dual Export: Built-in support for MP4 (requires ffmpeg) and GIF (via Pillow), complete with tqdm progress bars.

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How to Read the Plot (Visual Guide)

• The Log-Spiral Green (Background): A Base-3 polar coordinate system.
  Radius: logarithmic scale of the integer ($r = \log_3 n$)
  Angle: base-3 phase ($\theta = 2\pi{\log_3 n}$)

• White Starry Wells: Perfect powers of two ($2^N$).
  These represent the potential targets—or “cups”—densely scattered across the green.

• The Trajectory (Colored Path): The discrete steps of the Shiftless Collatz sequence.

• Friction Heatmap ($\mu_k$): The color and thickness of the path reflect the instantaneous LSB friction coefficient
  $$\mu_k = \frac{\mathrm{LSB}}{B_{k+1}}.$$

  • Cyan / Thin lines: Low friction. Historical momentum dominates, and the sequence moves smoothly along its deterministic inertia.
  • Magenta / Thick lines: High friction. Strong LSB intervention causes chaotic, sharp turns—representing critical “information collisions.”

• The Yellow Flag & Star: The final jackpot.
  The sequence reaches a pure power of two ($2^M$), fully resolving into the target well and ending the shot.

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Requirements

Install the required Python packages:

pip install numpy matplotlib tqdm

(Optional) For high-quality MP4 export, ensure ffmpeg is installed on your system.

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Usage

Run the script in your terminal or Jupyter Notebook.
You can change the starting number by modifying the seed variable:

# Change the seed to hunt different trajectories
seed = 27

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