Kakeya Needle Lab

An interactive desktop application for exploring the classical Kakeya Needle Problem through geometry, animation and mathematical visualization.

The application demonstrates how a unit-length needle can rotate through all possible directions while remaining inside different planar regions. It includes multiple geometric shapes, density visualization, endpoint trajectories, animated motion and export capabilities.

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Features

• Interactive needle animation
• Circle, Square, Triangle and Deltoid regions
• Endpoint trajectory visualization
• Needle center trajectory
• Swept density heatmap
• Perron Tree overlay
• Real-time geometric metrics
• Zoom and Pan
• High-resolution PNG export
• Animated GIF export
• Modular architecture

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Installation

Clone the repository

git clone https://github.com/your_username/kakeya-needle-lab.git

cd kakeya-needle-lab

Install dependencies

pip install -r requirements.txt

Run

python app.py

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Controls

Action	Description
Mouse Wheel	Zoom
Left Mouse Drag	Pan
Play	Start animation
Pause	Stop animation
Path Slider	Select animation position
Save PNG	Export current frame
Save GIF	Export full animation

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Visualization Options

• Needle
• Endpoints
• Center
• Boundary
• Region
• Grid
• Endpoint Paths
• Center Path
• Swept Density
• Heatmap
• Perron Tree Overlay

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Export

PNG

Export the current visualization as a high-resolution PNG image.

GIF

Render and export a complete 360° motion cycle as an animated GIF.

Configurable options include:

• Number of frames
• Frame duration
• Infinite looping

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Project Structure

kakeya-needle-lab/

├── app.py
├── export.py
├── geometry.py
├── raster.py
├── renderer.py
├── scene.py
├── ui.py
├── requirements.txt
│
├── motion/
│   ├── base.py
│   ├── circle.py
│   ├── deltoid.py
│   ├── factory.py
│   ├── polygon.py
│   ├── square.py
│   └── triangle.py
│
├── constructions/
│   └── perron.py
│
├── widgets/
│
└── tests/

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Mathematical Background

The Kakeya Needle Problem, proposed by Soichi Kakeya (1917), asks for the smallest planar region in which a unit-length needle can be continuously rotated through 360 degrees.

Subsequent work by A. S. Besicovitch showed that such sets may have arbitrarily small area, leading to one of the most fascinating topics in geometric measure theory.

This application focuses on interactive visualization of these geometric ideas rather than formal proofs.

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Technologies

• Python
• Tkinter
• NumPy
• Pillow
• Matplotlib

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Roadmap

Completed

• Interactive animation
• Multiple geometric regions
• Density visualization
• Perron Tree overlay
• PNG export
• GIF export
• Zoom & Pan

Future Ideas

• Besicovitch construction
• SVG export
• MP4 export
• Mathematical reports
• Additional Kakeya constructions

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Acknowledgements

This project was developed as an educational and visualization tool for exploring classical geometry, geometric motion and the Kakeya Needle Problem.