Fibonacci Spiral (approaches golden ratio)") plt.axis('equal') plt.show() def main(): parser = argparse.ArgumentParser(description='π and Golden Ratio Visualizations') parser.add_argument('--rectangle', action='store_true', help='Plot golden rectangle with arc using π') parser.add_argument('--spiral', action='store_true', help='Plot Fibonacci spiral') parser.add_argument('--phi', type=float, default=(1 + np.sqrt(5)) / 2, help='Golden ratio value (default φ)') parser.add_argument('--pi', type=float, default=np.pi, help='Pi value (default π)') parser.add_argument('--terms', type=int, default=10, help='Terms in Fibonacci spiral') args = parser.parse_args() if args.rectangle: golden_rectangle_plot(args.phi, args.pi) if args.spiral: fibonacci_spiral_plot(args.terms) if __name__ == "__main__": main() numpy matplotlib # π and Golden Ratio (φ) Exploration This repository demonstrates mathematical and geometric relationships between π (pi ≈ 3.14159) and the golden ratio (φ ≈ 1.61803) using Python visualizations. ## Features - **Golden Rectangle Visualization**: Plots a golden rectangle and overlays an arc using π. - **Fibonacci Spiral Visualization**: Plots a spiral built from Fibonacci numbers, which approximates the golden ratio. - **Command Line Input**: Select which visualizations to run and customize constants. ## Usage 1. Install requirements: 2. Plot a golden rectangle with a π arc: 3. Plot a Fibonacci spiral: 4. Customize parameters: ## Further Ideas - Add number theory demonstrations - Explore pentagons and other golden ratio shapes - Animate transitions or combine both visualizations ## License MIT License Here are resources with transcendence proofs: - A concise exposition for π: [Simple proofs: Pi is transcendental « Math Scholar](https://mathscholar.org/2025/02/simple-proofs-pi-is-transcendental/) - Mathematical StackExchange discussion and references: [Proof π is transcendental without symmetric function theory](https://math.stackexchange.com/questions/4861927/proof-pi-is-transcendental-without-symmetric-function-theory) - The Stacks Project on transcendence bases, including π: [Section 9.26: Transcendence—The Stacks project](https://stacks.math.columbia.edu/tag/030D) The golden ratio φ is **not transcendental** (it is algebraic: the root of x² − x − 1 = 0), so proofs of transcendence are only relevant for π and similar numbers. --- 1. [Simple proofs: Pi is transcendental « Math Scholar](https://mathscholar.org/2025/02/simple-proofs-pi-is-transcendental/) 2. [Proof π is transcendental without symmetric function theory](https://math.stackexchange.com/questions/4861927/proof-pi-is-transcendental-without-symmetric-function-theory) 3. [Section 9.26: Transcendence—The Stacks project](https://stacks.math.columbia.edu/tag/030D)

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